condensed matter - Why is the Fermi surface stable?

As a condensed matter physicist, I take it for granted that a Fermi surface is stable.

But it is stable with respect to what?

For instance, Cooper pairing is known as an instability of the Fermi surface.

I'm simply wondering what makes the Fermi surface stable?

Possible way of thinking: Is it a topological property of the Fermi gas (only of the free one ?, only robust against disorder?)? What is the modern, mathematical definition of the Fermi surface (shame on me, I don't even know this, and all my old textbooks are really sloppy about that, I feel)? What can destroy the Fermi surface, and what does destroy mean?

Any idea / reference / suggestion to improve the question is welcome.

Addenda / Other possible way to discuss the problem: After writing this question, I noted this answer by wsc, where (s)he presents a paper by M. Oshikawa (2000), Topological Approach to Luttinger’s Theorem and the Fermi Surface of a Kondo Lattice PRL 84, 3370–3373 (2000) (available freely on arXiv), and a paper by J. Luttinger & J. Ward Ground-State Energy of a Many-Fermion System. II. Phys. Rev. 118, 1417–1427 (1960). An other interesting reference to start with is a paper by J. Luttinger, Fermi Surface and Some Simple Equilibrium Properties of a System of Interacting Fermions, Phys. Rev. 119, 1153–1163 (1960), where he shows (eq.33) that the volume of the Fermi surface is conserved under interaction, using analytic properties of the Green function including the self-energy as long as the total number of particles is conserved. I'm not sure if it's sufficient to proof the stability of the Fermi surface (but what does stability means exactly, I'm now confused :-p ) Is there absolutely no modern (topological ?) version of this proof ?