Consider a closed orientable surface of negative Euler characteristic. In joint work with A. Katok, we showed the flexibility of metric and topological entropies of geodesic flow in the class of negatively curved metrics of fixed total area. In this talk, we will discuss the aforementioned result and the flexibility of entropies under the additional restriction that the metrics we consider are conformally equivalent to a fixed hyperbolic metric. It turns out that some restrictions arise in conformal classes. Also, we will point out some geometric consequences for those families of Riemannian metrics (joint with T. Barthelme).