It is well known that under appropriate geometric conditions, the solutions of the optimal transport (Monge-Kantorovich) problem can be found by solving a (degenerate) elliptic equation of Monge-Ampere type, which in turn can be found as the stationary solution of a parabolic equation. In this talk, I will discuss an exponential convergence result for solutions of such a parabolic equation. This is done by exploiting some of the hidden geometric flavor of the parabolic optimal transport equation, which ultimately connects to the pseudo-Riemannian metric for optimal transport first introduced by Kim and McCann. This talk is based on joint work with Farhan Abedin.