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The mathematical structure of thermodynamics by Peter Salamon (pdf) would be an example, but i would like a more abstract natural formulation of application of differential geometry or even geometric algebra to for example Maxwell relations in thermodynamics that does not use coordinates.

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    $\begingroup$ "or even geometric algebra"? $\endgroup$
    – Qfwfq
    Commented Apr 22, 2018 at 11:34

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some pointers to the literature:

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There is a calculus textbook which does this:

Bamberg and Sternberg A course in mathematics for students in physics, vol. II, Chap. 22, "Thermodynamics".

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In this context, one should probably mention "Contact Geometry: the Geometrical Method of Gibbs's Thermodynamics" by V.I. Arnold (Proc. of the Gibbs Symposium,Yale University, 1989, 163-179).

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