Something I’d always meant to do is learn something about thermodynamics. I’ve tried several times, but each time I get bogged down in the details of heat engines. This time, I found some very nice lecture notes by Nino Boccara. They provide an overview abstracted enough from the physical details to be informative to a mathematical audience. They take entropy as a primitive concept rather than deriving it from properties of heat engines.
(I’ve been trying to find other introductions from the same point of view of Boccara, but with more detail. I haven’t had any luck. Rather than sticking closely to ideal gases, Boccara takes a general view of the formalism, one that does not (in principle) even require energy as one of the state variables. His approach to statistical mechanics is related to the maximum entropy approach of E. T. Jaynes.)