  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 $\operatorname{RCD}$-spaces with
a negative lower Ricci curvature bound and Riemannian manifolds with a negative
$L^p$-integral Ricci curvature lower bound.
