From my own experience, I will advise you against every book of mathematical methods written specifically for physicist. From my point of view, is better to learn about mathematics from mathematically written books (it sounds so obvious but is not). For example, many people like Schultz, Geometrical methods of mathematical physics, but I prefer to learn about the common topics in Singer, Thorpe, Lecture notes on elementary topology and geometry. (I don't say it is not a good textbook, I only say I find difficult learning things on books written in a pretty informal way.)
The most complete work about methods of mathematical physics is probably
- Reed, Simon, Methods of modern mathematical physics,
that covers functional analysis, Fourier analysis, scattering theory, operator theory.
Since you are interested in cosmology, the best review on Loop Quantum Gravity is that by Thomas Thiemann,
- Thomas Thiemann, Modern and canonical quantum general relativity,
a 900 pages review, equipped with about 300 pages of mathematical methods (mathematical appendices are not a textbook however, but a collection of necessary results, eventually explored in some depth). References therein are very useful also.
Many people like
- Deligne et al., Quantum fields and strings: a course for mathematicians,
that joins a good part of your requests. (I haven't read it, however, I know it since is "famous".)
A celebrated book on methods of classical mechanics, concerning manifolds too, is
- Abraham, Mardsen, Foundations of mechanics.
Another is
- Choquet, Bruhat, Analysis, manifolds and physics.
All Arnold's books are always a great choice. (he wrote about ergodic theory and geometrical methods for differential equations, among the other things.)
There are a lots of more specific books, e.g. dealing with mathematical structure of quantum mechanics, but many of those are more and more specialized and is better to have very clear the general theory before try to get more involved into dangerous subjects such as, to say, quantum field theory. Once one has a strong background, the best opera on the subject of field theory probably is
- Zeidler, Quantum field theory,
an enormous amount of things (Zeidler style!) that covers all of the subject. Another excellent text on field theory is that of Haag,
- Haag, Local quantum physics.
EDIT. I'd like to add some book I've discovered more recently and I think fit very well:
Streater, Wightman, "PCT, Spin and all that",
Teschl, "Mathematical methods in Quantum Mechanics",
Bogolioubov, Logunov, Todorov, "Axiomatic Quantum Field Theory",
Lansdman, "Mathematical concepts between classical and quantum mechanics".
Best Answer
Thermodynamic formalism (TF) is a very exciting area which still to these days remains quite active. One of the most prominent applications of TF is fractal geometry. On of the most amazing results in my opinion, is Bowen's formula, which he derived to study the dimension of quasi-circles but it was then generalized to a more general setting. It essentially says that the dimension of certain fractals can be characterized as the unique solution to an equation involving a dynamically defined quantity (the pressure function, a central object of TF), which is also related to the spectrum of a certain class of dynamically defined operators. This allowed Ruelle to use some spectral theory to conclude that if you perturbate your fractals in a certain way, then their dimension varies very nicely (analytically in many cases). These notes are a good starting point to give you a more precise idea of what I am talking about:
http://www.mat.uc.cl/~giommi/notas3_escuela.pdf
If you want to go further into the details of these ideas, I highly recommend the book by Falconer, Fractal Geometry, where he goes into the details of the theory. Many of his works are also very important in establishing connections between TF and fractal geometry. His books are very nice to read also.
As you mentioned in one of the comments, the work of Pollicott is also very important in TF. He has a long history of trying to establish connections between TF and number theory, and lately he has been trying to find formulas for a circle packing problem:
https://arxiv.org/pdf/1704.06896.pdf
His works tend to be a bit harder to read though.
There are other uses of TF within the area of dynamical systems and ergodic theory, which concern the existence of invariant measures and equilibrium, which is a very big deal in general. The theory of Ruelle here is quite important, and you can read some nice notes by Walkden here:
https://personalpages.manchester.ac.uk/staff/charles.walkden/magic/lecture09.pdf
I think this should give you a good taste of different applications of TF.