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You can write a program general enough to be a universe but which doesn't involve temperature and doesn't involve inevitable information loss over time. Obviously none of them are going to be generating information from nowhere, but in principle it's at least possible to break even. (One example, which is rather simple and almost borders on cheating, would be to include an API that would allow any agent to access any bit of information from any point in the past. As far as I can tell, there's no reason why this wouldn't be allowed. It would have the aesthetic disadvantage of having a fundamentally directional time dimension, but that shouldn't cause any real problems to any agents living within it.)
Actually the lack of loss of information over time is precisely what generates the 2nd law of thermodynamics. Specifically, since all information from the past must thus be stored somewhere (unfortunately often in a way that's hard to access, e.g., the "random" motion of atoms) that continuously leaves less room for new information.
Right, sorry, I was referring to subjective information loss. I understand that information is globally conserved.
Won't help. You could have a universe that includes chronoscopes but still have the problem that it's continuously filling up with entropy.
Apparently I don't actually understand this subject, so I hereby relinquish my previous opinion about it and won't form a new one (beyond taking better-informed people's word for it for now) until I've learned it better.
You could have a universe though that gains more "room" for entropy faster than it gains entropy... so entropy keeps increasing, but there's an ever increasing entropy sink, right?
That's another way to do it.
One question one might ask is if our own universe has that property.