Entropy Rigidity and Hilbert Volume

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.