Equilibrium states in ergodic theory are defined via a variational principle where one takes the supremum over all invariant measures of the entropy + the average of the potential. This is therefore a Legendre transform in the general sense. In thermodynamics, Gibbs (equilibrium) states are instead defined by a Legendre transform assuming the differentiability of the average of the potential (and of the topological pressure). It seems to me that the formulation from ergodic theory is therefore more general since it does not assume a differentiable structure.
Under which assumptions on the underlying dynamical system does the supremum formulation from ergodic theory reduce to the differentiable formulation of thermodynamics ?
I guess that the two formulations coincide at least in one case, i.e. whenever all microstates have the same probability. Physically, this is what happens for steady solutions of Liouville equation; it leads both to maximisation of the familiar, textbook entropy and to a configuration (thermodynamical equilbrium) where ergodicity is satisfied. But I understand your question has a much more general outlook. As far as I know, given a suitably general definition of 'entropy' it is often possible to find Legendre transform in the most bizarre applications, including even the Schroedinger equation of quantum mechanics (see e.g. S. P. Flego, A. Plastino and A. R. Plastino, Fisher Information and Quantum Mechanics, International Research Journal of Pure & Applied Chemistry, 2(1): 25-54, 2012, www.sciencedomain.org). Then, I suppose that the answer to your question depends critically on the detailed structure of the variational principle you are referring to.
The ergodic theoretic formulation is for discrete time dynamical systems in its most well known formulation. It is often formulated on subshifts which are associated to the geometric model via the construction of so-called Markov partitions. Try to look at the book of Zinsmeister on Thermodynamical formalism, I hope it helps.
Entropy: A concept that is not a physical quantity
https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity
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