Strictly non-proportional geodesically equivalent metrics have htop(g) = 0

If a closed manifold M possesses two Riemannian metrics which have the same unparameterized geodesics and are not strictly proportional at each point, then the topological entropy of both geodesic flows is zero. This is the main result of the paper and it has many dynamical and topological corollaries. In particular, such a manifold M should be finitely covered by the product of a rationally elliptic manifold and a torus.