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The volume entropy of a compact metric measure space is known to be theexponential growth rate of the measure lifted to its universal cover atinfinity. For a compact Riemannian $n$-manifold with a negative lower Riccicurvature bound and a upper diameter bound, it was known that it admits analmost maximal volume entropy if and only if it is diffeomorphic andGromov-Hausdorff close to a hyperbolic space form. We prove the quantitativerigidity of almost maximal volume entropy for $\operatorname{RCD}$-spaces witha negative lower Ricci curvature bound and Riemannian manifolds with a negative$L^p$-integral Ricci curvature lower bound.

Quantitative rigidity of almost maximal volume entropy for both RCD spaces and integral Ricci curvature bound