  For a closed, strictly convex projective manifold of dimension $n\geq 3$ that
admits a hyperbolic structure, we show that the ratio of Hilbert volume to
hyperbolic volume is bounded below by a constant that depends only on
dimension. We also show that for such spaces, if topological entropy of the
geodesic flow goes to zero, the volume must go to infinity. These results
follow from adapting Besson--Courtois--Gallot's entropy rigidity result to
Hilbert geometries.
