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We will show that the quantitative maximal volume entropy rigidity holds onAlexandrov spaces. More precisely, given $N, D$, there exists $\epsilon(N,D)>0$, such that for $\epsilon<\epsilon(N, D)$, if $X$ is an $N$-dimensionalAlexandrov space with curvature $\geq -1$, $\operatorname{diam}(X)\leq D,h(X)\geq N-1-\epsilon$, then $X$ is Gromov-Hausdorff close to a hyperbolicmanifold. This result extends the quantitive maximal volume entropy rigidity of\cite{CRX} to Alexandrov spaces. And we will also give a quantitative maximalvolume entropy rigidity for $\op{RCD}^*$-spaces in the non-collapsing case.

Quantitative maximal volume entropy rigidity on Alexandrov spaces