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The volume entropy of a compact metric measure space is known to be the exponential growth rate of the measure lifted to its universal cover at infinity. For a compact Riemannian n -manifold with a negative lower Ricci curvature bound and a upper diameter bound, it was known that it admits an almost maximal volume entropy if and only if it is diffeomorphic and Gromov-Hausdorff close to a hyperbolic space form. We prove the quantitative rigidity of almost maximal volume entropy for RCD ⁎ -spaces with a negative lower Ricci curvature bound and Riemannian manifolds with a negative L p -integral Ricci curvature lower bound. [ABSTRACT FROM AUTHOR]