Gauss-Bonnet is used extensively in the proofs of “no-hair” and “positive mass”:
Israel, Werner, Event Horizons in Static Vacuum Space-Times, Phys. Rev. 164, 1776-1779 (1967).
Schoen, Richard; Yau, Shing-Tung, On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys. 65, 45-76 (1979). ZBL0405.53045.