References about rigorous thermodynamics
The pioneer of the rigorous treatment of thermodynamics is Constantin Carathéodory. His article (Carathéodory, C., Untersuchung über die Grundlagen der Thermodynamik, Math. Annalen 67, 355-386) is cited everywhere in this context, but probably you want some newer and more modern things.
Buchdahl wrote a lot of papers about this subject in the 40's, 50's and 60's. He summarized these in the book: H.A. Buchdahl, The Concepts of Classical Thermodynamics (Cambridge Monographs on Physics), 1966.
There was a recent series of articles on this subject by Lieb and Yngavason which became famous. You can find the online version of these here, here, here and here :).
Finally, I have come across the book T. Matolcsi, "Ordinary Thermodynamics" (since a few friends of mine went to the class of the author), which treats thermodynamics in a mathematically very rigorous way.
I hope some of these will help you.
I think this book is what you are looking for: (although this question is already answered a long time ago)
Thermodynamics and an Introduction to Thermostatistics, by Herbert B. Callen
It is sort of a modern version of the Gibbs formulation. Entropy is postulated at the beginning, from which all Thermodynamics is formally derived. The scheme is somewhat abstract, but the book is easy to read. It points in the direction of statistical mechanics (specially near the end of the book).
As some people point out, it depends on what you mean by "rigorous". If you mean rigorous, in the sense of a mathematician the books by David Ruelle are very nice.
- Statistical mechanics: rigorous results by David Ruelle. This is actually a very nice books, which takes the statistical mechanics point of view (instead of a phenomenological/geometric approach). Everything is defined very rigorously, the existence of the thermodynamic limit is proven in both classical/quantum and lattice/continuous systems (for well-behaved interactions) and the equivalence of different ensembles shown. Also phase transitions are covered where very general versions of the Lee-Yang and the Mermin-Wagner theorems are proven. And much more. The book can be rather dry and formal, so make sure to find the old papers by Michael Fisher which can be helpful.
- Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics by David Ruelle. I don't know this book very well, but it is also a very rigorous (and more advanced than 1.) treatment of thermodynamics.
Hope you will find this useful.