I will discuss two notions of entropy for the geodesic flow: the topological entropy and the measure-theoretic entropy with respect to the Liouville measure. How these dynamical invariants relate to the underlying Riemannian metric has long been of interest. For negatively curved surfaces, Katok proved that equality of the Liouville and topological entropies characterizes metrics of constant negative curvature. In this setting, Manning asked how these two entropies vary along the normalized Ricci flow; more specifically, he proved the topological entropy is monotonic. The main result of this talk, joint with Erchenko, Humbert, and Mitsutani, is that the same is true for the Liouville entropy. In addition to geometric and dynamical methods, we use microlocal analysis to differentiate the Liouville entropy with respect to a conformal perturbation of the metric.