I think the naive expectation would be that a hot state would be highly entangled and non-separable. The non-separability is what leads to decoherence when you do a partial trace. You can arbitrarily split any multi-part Hilbert space into a "system" and an "environment" and then trace out the environment. If they are highy entangled with one another then the resulting system-without-environment will be decohered.

Alternatively formulated, this suggests that an entangled state, if weakly coupled to a bath at high temperature and allowed to thermalize, will become fully disentangled at a finite time. (p1)

hinting that something different is happening here (from the C&L model mentioned below.)

Indeed the CS guys are probably looking for a way to redefine entanglement (entropy, as you summarized above), to essentially “work” in the same way as what they have in their preprint, but also cover the prethermalized cases..

Hmm, C&L argument involves taking a classical limit, does it not? This paper seems “fully quantum” (provided that you agree that the maximally mixed “Gibbs” state should not be considered a “classical” limit, only an infinite temp limit)

Yeah maybe that's what's bothering me, what is the high temperature limit if not the classical limit?

Heh, you might be additionally bothered that for the C&L model, the strong coupling (to heat bath) limit can also be considered the (/a?) classical limit, e.g.

https://arxiv.org/pdf/1603.07495