In this talk we will discuss the Besson-Courtois-Gallot (BCG) theorem in the context of convex projective geometry. The BCG theorem is a rigidity statement relating the volume and entropy of a negatively curved Riemannian manifold, and has many applications including Mostow rigidity. In the world of convex real projective structures, the natural Hilbert geometry on these objects is only Finsler and the geometry is generally not even $C^2$. We discuss our analogous BCG theorem and some applications in the case where the manifold is closed. We will include some ongoing work to extend the result to finite volume. This is based on joint work with Ilesanmi Adeboye and David Constantine.