> what happens to time as you asymptotically approach pure entropy

Uh, maybe nothing? Depends on how you approach the pure entropy condition, and where the observer is and how it's moving compared to a clock.

Let's consider the measure of entropy in Boltzmann's relation : S = k log W or essentially that the quantity of entropy is the log relationship between the macrostate and the number of possible microstates that can represent that macrostate. A tree is a macrostate. Its individual cells, or the atoms that make those cells, or the quarks that make the nuclei, are examples of ever more micro microstates.

You are pretty low entropy because we can't take a cubic centimetre of one of your bones and swap it with a cubic centimetre around the mitral valve of your heart and expect the same macrostate to persist: you rapidly become a corpse. Empty space is "pure" entropy because we can take a cubic centimetre of vacuum and swap it with a cubic centimetre of vacuum elsewhere, and it will make absolutely no difference to the macrostate. We can swap all of the cubic centimetres of vacuum in an empty spacetime with one another, so the combinations at just that scale that produce the same empty spacetime is enormous. A black hole (assuming no-hair conjecture is true) can be represented by eleven variables, most of which vanish by a suitable choice of coordinates, leaving mass as the dominating item in the macrostate. But that mass can be all hydrogen atoms, or half as many helium atoms, or some arbitrary mix of gas and dust, but those all don't matter to the horizon that defines the black hole. The entropy of the horizon is enormous, because we can't distinguish between an isolated black hole made of nothing but atomic hydrogen to a different isolated black hole of the same mass, charge, and spin made of nothing but molecular hydrogen.

Now let's add a clock somewhere in these vast seas of entropy. How fast does it tick? It depends on who is looking! If we put it into otherwise empty expanding space, someone close to it will see it ticking faster than someone extremely far away from it. The latter, watching for long enough, will see the clock slow, dim, shrink, and ultimately vanish thanks to the metric expansion of space. The former, staying close by the clock, would see it tick at the same rate for an arbitrarily long time. But if we put it close to a black hole, what happens? The close by observer, staying close to it, will see it tick at the same rate for an arbitrarily long time. Observers at greater distance will see it ticking more slowly. If it were to fall into a very large black hole along with the close by observer, there is "no drama": the observer sees the same ticking rate upon them crossing the event horizon together. But the distant observer will see the clock slow, dim, and shrink until it vanishes. (Don't be tempted to think that a black hole is the same as an expanding universe because of this coincidence! We are definitely not inside a black hole, but we are definitely inside an expanding universe. The metrics are very different and lead to different trajectories for things moving through the respective spacetimes, even though the very different spacetimes coincidentally have boundaries that can in some circumstances produce similar observables for a clock near the boundary.)

We can complicate the picture by having the observer move relativistically with respect to the clock, especially if we introduce extreme acceleration.

So the thing about time in a relativistic context: how fast a clock ticks is observer-dependent, and in curved spacetime it's position-dependent too. It's the curvature of spacetime that makes the context relativistic; from the equivalence principle (non-gravitational) acceleration likewise makes the context relativistic.

If we lower the entropy of the vacuum by introducing things like galaxy clusters and cats, these less-entropic things generate curvature. So there are more places for a clock to slow down in a universe that has less entropy, than in a universe which has more entropy. Even in the black hole case, we can add another black hole and now there are two event horizons around which a clock will tick slowly from the perspective of a distant observer. Add more black holes, add more places where the clock could be ticking slowly for this observer. Of course, as you add more black holes, you are generating a more complicated macrostate, so entropy is dropping.

So the answer is: as a universe's entropy increases, there are fewer places for clocks to run slowly.

Even though a uniform gas has very high entropy, it still has less entropy than empty vacuum, and it is also subject to things like brownian motion and other fluctuations in density. Is a Boltzmann brain inevitable in such a gas (in the farrrrrr future of our universe, there may be a sparse gas of ultra-infrared photons with an even sparser gas of black holes and diffuse ordinary matter, and it may be uniform after a verrrry long time) ? No, but we can assign a low probability that the matter will fluctuate into a Boltzmann brain. The problem that obsesses Boltzmann brain cosmologists is that the low probability of a Boltzmann brain with false memories of having grown up in a universe like ours is enormously enormously enormously more likely than the ultra-low entropy of a hot big bang which evolves into a "real" brain with "real" memories of having grown up with cats and computers and so on, with entropy increasing as you get further from the big bang[1].

So does that mean you are a Boltzmann brain with false memories?

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[1] The early universe is almost certainly low entropy. If, when it is suuuuuper dense but with tiny variations, we swap some of it at "A" with some of it at "B", the stuff around "A" won't be dense enough to collapse into the galaxy it otherwise would have. Or maybe the stuff at "B" will suddenly be at a critical density to start collapsing gravitationally, leading to star formation and so on. The tiny variations are expected because of quantum behaviour at densities where quantum gravitation is important. Also, of course, it would be weird if the second law of thermodynamics depends on how close to the early universe you are: and that would be the case if the early universe had more entropy than the present universe (which is full of lots of nearly empty space, with a large and growing number of indistinguishable cubic centimetres between galaxy clusters).