$C^0$-Robustness of topological entropy for geodesic flows

In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the $C^0$ topology. We establish several instances of entropy robustness (persistence of entropy non-vanishing after small $C^0$ perturbations). A large part of this paper is dedicated to metrics on the 2-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy (thus generalizing and quantifying a result of Denvir-Mackay) and that metrics with robust positive entropy on the torus are $C^{\infty}$ generic. Moreover, we quantify the asymptotic behavior of volume entropy in the Teichm\~A{\OE}ller space of hyperbolic metrics on a punctured torus, which bounds from below the topological entropy for these metrics. For general closed manifolds of dimension at least 2 we prove that the set of metrics with robust and high positive entropy is $C^0$-large in the sense that it is dense, contains cones and arbitrarily large balls.