Maximal volume entropy rigidity for RCD*(-(N-1),N) spaces

For n-dimensional Riemannian manifolds M with Ricci curvature bounded below by -(n-1), the volume entropy is bounded above by n-1. If M is compact, it is known that the equality holds if and only if M is hyperbolic. We extend this result to RCD*(-(N-1),N) spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in RCD* spaces. As an application, we obtain an almost rigidity result which partially recovers a result by Chen-Rong-Xu for Riemannian manifolds.