Strictly non-proportional geodesically equivalent metrics have   $h_\text{top}(g)=0$
  Suppose the Riemannian metrics $g$ and $\bar g$ on a closed connected manifold $M^n$ are geodesically equivalent and strictly non-proportional at least at one point. Then the topological entropy of the geodesic flow of $g$ vanishes. 
Boris S. Kruglikov and Vladimir S. Matveev