What I know (and please correct me if I'm stating malarkey): the entropy of the universe (its description) is contained in Weyl tensor. Einstein's field equations don't directly relate the entropy with the universe and its curvature/geometry. We can obtain Weyl tensor by contracting the Riemann tensor which in $4$D has only $20$ independent components (and Ricci tensor has $10$).
In Cosmology, to describe the evolution of the Universe, Einstein equations are not enough (indeed we need Friedmann Equations too and else), hence to understand the evolution what we do is to look at the independent components of the Weyl tensor.
But when one comes to deal with Big Bang, then Weyl tensor vanishes, whereas they become larger and larger the more the Universe expands. This could be the explanation of why the entropy does always increase (at least without starting to run exotic physics and so on).
Now, the Universe is not a closed system, and it cannot be described by usual Thermodynamics because it has no volume and no temperature, and we cannot run experiments in the thermodynamics sense to study it in that way. Hence to speak about Entropy in the Clausius sense we need to consider it as a sum of portions (read: closed systems) and look at the interactions in the neighborhood.
Question: What happens to the curvature of the Universe if the Entropy of the universe wouldn't conserve? Is the increasing / conservation / non-conservation of the Entropy, related to something like the the energy density of the Universe? Maybe it would be comparable to the critical energy density?