Computing near the Sun costs more because it's hotter, sure. Fortunately, I understand that the Sun produces hundreds, even thousands of times more energy than a little fusion reactor does, so some inefficiencies are not a problem.
Every practical computational tech substrate has some error bounded compute/temperature curve, where computational capability quickly falls to zero past some upper bound temperature. Even for our current tech, computational capacity essentially falls off a cliff somewhere well below 1,000K.
My general point is that the really advanced computing tech shifts all those curves over - towards lower temperatures. This is a hard limit of physics, it can not be overcome. So for a really advanced reversible quantum computer that employs superconduction and long coherence quantum entanglement, 1K is just as impossible as 1,000K. It's not entirely a matter of efficiency.
Another way of looking at it - advanced tech just requires lower temperatures - as temperature is just a measure of entropy (undesired/unmodeled state transitions). Temperature is literally an inverse measure of computational potential. The ultimate computer necessarily must have a temperature of zero.
You say that the reversible brains don't need that much energy.
At the limits they need zero. Approaching anything close to those limits they have no need of stars. Not only that, but they couldn't survive any energy influx much larger than some limit, and that limit necessarily must go to zero as their computational capacity approaches theoretical limits.
If it's energy, then they will want to pipe in as much energy as possible from their local star.
No. There is an exact correct amount of energy to pipe in based on their viable operating temperature of their current tech civ. And this amount goes to zero as you advance up the tech.
It may help to consider applying your statement to our current planet civ. What if we could pipe in 10000x more energy than we currently receive from the sun. Wouldn't that be great? No. It would cook the earth.
The same principle applies, but as you advance up the ultra-tech ladder, the temp ranges get lower and lower (because remember, temp is literally an inverse measure of maximum computational capabillity).
OK, but more computing power is always better, the cold brains want as much as possible, so what limits them?
Given some lump of matter, there is of course a maximum information storage capacity and a max compute rate - in a reversible computer the compute rate is bounded by the maximum energy density the system can structurally support which is just bounded by its mass. In terms of ultimate limits, it really depends on whether exotic options like creating new universes are practical or not. If creating new universes is feasible, there probably are no hard limits, all limits becomes soft.
So you should get a universe of Dyson spheres feeding out mass-energy to the surrounding cold brains who are constantly colonizing fresh systems for more mass-energy to compute in the voids with
Dyson spheres are extremely unlikely to be economically viable/useful, given the low value of energy past a certain tech level (vastly lower energy need per unit mass).
Cold brains need some mass, the question then is how the colonization value of mass varies across space. Mass that is too close to a star would need to be moved away from the star, which is very expensive.
So the most valuable mass that gets colonized first would be the rogue planets/nomads - which apparently are more common than attached planets.
If colonization continues long enough, it will spread to lower and lower valued real estate. So eventually smaller rocky bodies in the outer system get stripped away, slowly progressing inward.
The big unknown variable is again what the end of tech in the universe looks like, which gets back to that new universe creation question. If that kind of ultimate/magic tech is possible, civs will invest everything in to that, and you have less colonization, depending on the difficulty/engineering tradeoffs.
Given some lump of matter, there is of course a maximum information storage capacity and a max compute rate - in a reversible computer the compute rate is bounded by the maximum energy density the system can structurally support which is just bounded by its mass. In terms of ultimate limits, it really depends on whether exotic options like creating new universes are practical or not. If creating new universes is feasible, there probably are no hard limits, all limits becomes soft.
This still doesn't answer my question. I understand your points about why ...
Our sun appears to be a typical star: unremarkable in age, composition, galactic orbit, or even in its possession of many planets. Billions of other stars in the milky way have similar general parameters and orbits that place them in the galactic habitable zone. Extrapolations of recent expolanet surveys reveal that most stars have planets, removing yet another potential unique dimension for a great filter in the past.
According to Google, there are 20 billion earth like planets in the Galaxy.
A paradox indicates a flaw in our reasoning or our knowledge, which upon resolution, may cause some large update in our beliefs.
Ideally we could resolve this through massive multiscale monte carlo computer simulations to approximate Solonomoff Induction on our current observational data. If we survive and create superintelligence, we will probably do just that.
In the meantime, we are limited to constrained simulations, fermi estimates, and other shortcuts to approximate the ideal bayesian inference.
The Past
While there is still obvious uncertainty concerning the likelihood of the series of transitions along the path from the formation of an earth-like planet around a sol-like star up to an early tech civilization, the general direction of the recent evidence flow favours a strong Mediocrity Principle.
Here are a few highlight developments from the last few decades relating to an early filter:
The Future(s)
When modelling the future development of civilization, we must recognize that the future is a vast cloud of uncertainty compared to the past. The best approach is to focus on the most key general features of future postbiological civilizations, categorize the full space of models, and then update on our observations to determine what ranges of the parameter space are excluded and which regions remain open.
An abridged taxonomy of future civilization trajectories :
Collapse/Extinction:
Civilization is wiped out due to an existential catastrophe that sterilizes the planet sufficient enough to kill most large multicellular organisms, essentially resetting the evolutionary clock by a billion years. Given the potential dangers of nanotech/AI/nuclear weapons - and then aliens, I believe this possibility is significant - ie in the 1% to 50% range.
Biological/Mixed Civilization:
This is the old-skool sci-fi scenario. Humans or our biological descendants expand into space. AI is developed but limited to human intelligence, like CP30. No or limited uploading.
This leads eventually to slow colonization, terraforming, perhaps eventually dyson spheres etc.
This scenario is almost not worth mentioning: prior < 1%. Unfortunately SETI in current form is till predicated on a world model that assigns a high prior to these futures.
PostBiological Warm-tech AI Civilization:
This is Kurzweil/Moravec's sci-fi scenario. Humans become postbiological, merging with AI through uploading. We become a computational civilization that then spreads out some fraction of the speed of light to turn the galaxy into computronium. This particular scenario is based on the assumption that energy is a key constraint, and that civilizations are essentially stellavores which harvest the energy of stars.
One of the very few reasonable assumptions we can make about any superintelligent postbiological civilization is that higher intelligence involves increased computational efficiency. Advanced civs will upgrade into physical configurations that maximize computation capabilities given the local resources.
Thus to understand the physical form of future civs, we need to understand the physical limits of computation.
One key constraint is the Landauer Limit, which states that the erasure (or cloning) of one bit of information requires a minimum of kTln2 joules. At room temperature (293 K), this corresponds to a minimum of 0.017 eV to erase one bit. Minimum is however the keyword here, as according to the principle, the probability of the erasure succeeding is only 50% at the limit. Reliable erasure requires some multiple of the minimal expenditure - a reasonable estimate being about 100kT or 1eV as the minimum for bit erasures at today's levels of reliability.
Now, the second key consideration is that Landauer's Limit does not include the cost of interconnect, which is already now dominating the energy cost in modern computing. Just moving bits around dissipates energy.
Moore's Law is approaching its asymptotic end in a decade or so due to these hard physical energy constraints and the related miniaturization limits.
I assign a prior to the warm-tech scenario that is about the same as my estimate of the probability that the more advanced cold-tech (reversible quantum computing, described next) is impossible: < 10%.
From Warm-tech to Cold-tech
There is a way forward to vastly increased energy efficiency, but it requires reversible computing (to increase the ratio of computations per bit erasures), and full superconducting to reduce the interconnect loss down to near zero.
The path to enormously more powerful computational systems necessarily involves transitioning to very low temperatures, and the lower the better, for several key reasons:
Assuming large scale quantum computing is possible, then the ultimate computer is thus a reversible massively entangled quantum device operating at absolute zero. Unfortunately, such a device would be delicate to a degree that is hard to imagine - even a single misplaced high energy particle could cause enormous damage.
Stellar Escape Trajectories
The Great Game
If two civs both discover each other's locations around the same time, then MAD (mutually assured destruction) dynamics takeover and cooperation has stronger benefits. The vast distances involve suggest that one sided discoveries are more likely.
Spheres of Influence
Conditioning on our Observational Data
Observational Selection Effects
All advanced civs will have strong instrumental reasons to employ deep simulations to understand and model developmental trajectories for the galaxy as a whole and for civilizations in particular. A very likely consequence is the production of large numbers of simulated conscious observers, ala the Simulation Argument. Universes with the more advanced low temperature reversible/quantum computing civilizations will tend to produce many more simulated observer moments and are thus intrinsically more likely than one would otherwise expect - perhaps massively so.
Rogue Planets
Although the error range is still large, it appears that free floating planets outnumber planets bound to stars, and perhaps by a rather large margin.
Assuming the galaxy is colonized: It could be that rogue planets form naturally outside of stars and then are colonized. It could be they form around stars and then are ejected naturally (and colonized). Artificial ejection - even if true - may be a rare event. Or not. But at least a few of these options could potentially be differentiated with future observations - for example if we find an interesting discrepancy in the rogue planet distribution predicted by simulations (which obviously do not yet include aliens!) and actual observations.
Also: if rogue planets outnumber stars by a large margin, then it follows that rogue planet flybys are more common in proportion.
Conclusion
SETI to date allows us to exclude some regions of the parameter space for alien civs, but the regions excluded correspond to low prior probability models anyway, based on the postbiological perspective on the future of life. The most interesting regions of the parameter space probably involve advanced stealthy aliens in the form of small compact cold objects floating in the interstellar medium.
The upcoming WFIST telescope should shed more light on dark matter and enhance our microlensing detection abilities significantly. Sadly, it's planned launch date isn't until 2024. Space development is slow.