In the first part of my talk, I will review basic aspects of the so-called Casini-Huerta-Myers prescription employed to holographically compute Rényi entropies. The second part of the talk is focused on the analysis of the thermodynamics of black holes with hyperbolic event horizon in gravity theories with higher derivative corrections (Lovelock gravity theories). These black holes are indeed unstable, and, in the last part of the talk, I will discuss the implications of this instability for Rényi entropies.