These were good questions.

> Brownian motion which is for all intents and purposes time-reversible

How does the Brownian motion in a hot cup of tea compare to the Brownian motion in a room-temperature cup of tea? If you wait long enough, how does your hot tea not cool down?

(Or you can substitute an ideal gas in a container of volume V and the same ideal gas after adiabatic cooling after one increases V to say 4V while keeping everything else the same.)

> does the time dimension of the universe disappear

On theoretical grounds, no. All timelike and lightlike trajectories (that do not enter black holes) extend to the infinite future.

On practical grounds, maybe. In the standard model, our universe's volume keeps expanding into the infinite future, adiabatically cooling the matter (and radiation) within it, leading to among other things the redshifting of the cosmic microwave background. There is also cold (and cooling) radiation from the cosmological horizon. These photons may become very difficult, and in practice impossible, for some future scientist not accelerated with respect to the cosmic horizon to detect. However, strong acceleration, or a boost with respect to the cosmic frame, gives hotter and more particles in one direction, making them easier to detect.

So if even in the extreeeeeeeme future there exists the possibility of an accelerated or ultraboosted observer, can one say that the universe is truly frozen up to undetectaby small fluctuations in the energy-density of a (set of) chosen point(s)?

There are certainly ultraboosted observers of a sort now: cosmic rays ejected from extreme events like blazars and supernovae, including the famous "oh my god!" particle are examples. The GZK cutoff is a mechanism by which ultraboosted charged particles interact (given sufficiently long distances) with very low frequency photons, such as those from the cosmic microwave background. This mechanism should persist into the infinite future, so ultra high energy particles spit out of the last evaporating black holes might create interesting fireworks that way, for a very long time. They'd become rarer and rarer, but it's not clear they would ever totally vanish (it's something I'd have to actually calculate because I don't trust my intuition there!). Even if such naturally ultraboosted photons did become extinct, a future observer with an enormous particle accelerator could probably scatter charged particles off distant and very red photons. It might take a long time for the ultraboosted particle to interact with a very red photon and thus the roundtrip might be annoying for that future scientist, but at least there is a way to tell that the universe beyond its local laboratory is not completely empty (indeed with some care the scientist could continue to track the expansion history of the very red and very sparse universe of many trillions of years from now).

> highest state

The highest Boltzmann entropy would be complete vacuum, but that is unachievable because of horizon radiation, and therefore extreme observers (extremes of linear momentum against the horizons and the cosmic microwave background; extremes of spin angular momentum; extremes of acceleration) can still get a fairly exciting sky even as the universe empties out. They in turn might also produce exciting moments for bored observers of a very very dark sky.

Penultimately, "sky" and the equivalence principle: if there remains a planet with a high surface gravity after galaxies are gone and black holes have evaporated, any instrument placed on that will get a view with hotter/bluer and more particles than an low-mass instrument (like JWST mass) freely floating in space and seeing no (or only very very weak) dipole isotropy to the CMB.

And finally, that's not super fanciful - we already know some stars have been ejected from their galaxies (and very likely galaxy clusters), so a "dead" planet in its own Hubble volume is plausible. Maybe an observatory was built on a superearth before it escaped into inter-galaxy-cluster space, equipped with some reservoir of power that can last until the dark times.

So,

> entropy maximum

Given current knowledge, totally empty space is not a configuration our universe will relax into, although it'll be a very good approximation.

> time dimension

Here we return to the theory. Our universe is locally Lorentz-invariant, meaning there are three spacelike dimensions and one timelike one, and physics doesn't change for a system displaced very slightly in space or in time. There is no way to change that aspect of the universe without a fairly dramatic modification of extremely well tested parts of modern physics, and that necessarily includes a cut-off mechanism that keeps what we see here-and-now (and older distant things) locally Lorentz-invariant. (The other end of the universe has this issue too: if you want different dimensions or want break the translation invariance in our 3+1 dimensions near the hot big bang you have to make them something we don't notice here-and-now or in faraway things we see that are backlit by distant quasars).

Local Lorentz invariance everywhere leads to my paragraph near the top. It gives us three types of trajectory (spacelike, timelike, lightlike) and guarantees that the latter two types all go to the infinite future (if they do not enter black holes). <https://en.wikipedia.org/wiki/Causal_structure#Tangent_vecto...> for details.