  In this paper we provide two new characterizations of real hyperbolic
$n$-space using the Poincar\'e exponent of a discrete group and the volume
growth entropy. The first characterization is in the space of Hilbert metrics
and generalizes a result of Crampon. The second is in the space of Riemannian
metrics with Ricci curvature bounded below and generalizes a result of
Ledrappier and Wang.
