We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given
N
,
D
N, D
, there exists
ϵ
(
N
,
D
)
>
0
\epsilon (N, D)>0
, such that for
ϵ
>
ϵ
(
N
,
D
)
\epsilon >\epsilon (N, D)
, if
X
X
is an
N
N
-dimensional Alexandrov space with curvature
≥
−
1
\geq -1
,
diam
(
X
)
≤
D
,
h
(
X
)
≥
N
−
1
−
ϵ
\operatorname {diam}(X)\leq D, h(X)\geq N-1-\epsilon
, then
X
X
is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity provided by Chen, Rong, and Xu [J. Differential Geom. 113 (2019), pp. 227–272] to Alexandrov spaces. And we will also give a quantitative maximal volume entropy rigidity for
RCD
∗
\operatorname {RCD}^*
-spaces in the non-collapsing case.