  Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic
K\"ahler manifold $(Y,g)$ and a compact complex {hyperbolic manifold}
$(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of
$(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$.
When the lower bound is attained we get geometric rigidity theorems for the
diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and
S. Gallot [2] for the volume entropy. As a corollary, when $X=Y$, we show that
the minimal diastatic entropy is achieved if and only if $g$ is holomorphically
or anti-holomorphically isometric to the hyperbolic metric $g_0$.
