Imagine that the universe is approximately as it appears to be (I know, this is a controversial proposition, but bear with me!). Further imagine that the many worlds interpretation of Quantum mechanics is true (I'm really moving out of Less Wrong's comfort zone here, aren't I?).

Now assume that our universe is in a situation of false vacuum - the universe is not in its lowest energy configuration. Somewhere, at some point, our universe may tunnel into true vacuum, resulting in a expanding bubble of destruction that will eat the entire universe at high speed, destroying all matter and life. In many worlds, such a collapse need not be terminal: life could go one on a branch of lower measure. In fact, anthropically, life will go on somewhere, no matter how unstable the false vacuum is.

So now assume that the false vacuum we're in is highly unstable - the measure of the branch in which our universe survives goes down by a factor of a trillion every second. We only exist because we're in the branch of measure a trillionth of a trillionth of a trillionth of... all the way back to the Big Bang.

None of these assumptions make any difference to what we'd expect to see observationally: only a good enough theory can say that they're right or wrong. You may notice that this setup transforms the whole universe into a quantum suicide situation.

The question is, how do you go about maximising expected utility in this situation? I can think of a few different approaches:

1. Gnaw on the bullet: take the quantum measure as a probability. This means that you now have a discount factor of a trillion every second. You have to rush out and get/do all the good stuff as fast as possible: a delay of a second costs you a reduction in utility of a trillion. If you are a negative utilitarian, you also have to rush to minimise the bad stuff, but you can also take comfort in the fact that the potential for negative utility across the universe is going down fast.
2. Use relative measures: care about the relative proportion of good worlds versus bad worlds, while assigning zero to those worlds where the vacuum has collapsed. This requires a natural zero to make sense, and can be seen as quite arbitrary: what would you do about entangled worlds, or about the non-zero probability that the vacuum-collapsed worlds may have worthwhile life in them? Would the relative measure user also put zero value to worlds that were empty of life for other reasons than vacuum collapse? For instance, would they  be in favour of programming an AI's friendliness using random quantum bits, if it could be reassured that if friendliness fails, the AI would kill everyone immediately?
3. Deny the measure: construct a meta ethical theory where only classical probabilities (or classical uncertainties) count as probabilities. Quantum measures do not: you care about the sum total of all branches of the universe. Universes in which the photon went through the top slit, went through the bottom slit, or was in an entangled state that went through both slits... to you, there are three completely separate universes, and you can assign totally unrelated utilities to each one. This seems quite arbitrary, though: how are you going to construct these preferences across the whole of the quantum universe, when forged your current preferences on a single branch?
4. Cheat: note that nothing in life is certain. Even if we have the strongest evidence imaginable about vacuum collapse, there's always a tiny chance that the evidence is wrong. After a few seconds, that probability will be dwarfed by the discount factor of the collapsing universe. So go about your business as usual, knowing that most of the measure/probability mass remains in the non-collapsing universe. This can get tricky if, for instance the vacuum collapsed more slowly that a factor of a trillion a second. Would you be in a situation where you should behave as if you believed vacuum collapse for another decade, say, and then switch to a behaviour that assumed non-collapse afterwards? Also, would you take seemingly stupid bets, like bets at a trillion trillion trillion to one that the next piece of evidence will show no collapse (if you lose, you're likely in the low measure universe anyway, so the loss is minute)?