Note on stability of new hyperbolic AdS black holes and phase transitions in R\'{e}nyi entropies

We construct a series of new hyperbolic black hole solutions in Einstein-Scalar system and we apply holographic approach to investigate the spherical R\'{e}nyi entropy in various deformations of dual conformal field theories (CFTs). Especially, we introduce various powers of scalars in the scalar potentials for massive and massless scalar. These scalar potentials correspond to deformation of dual CFTs. Then we solve asymptotically hyperbolic AdS black hole solutions numerically. We map the instabilities of these black hole solutions to phase transitions of field theory in terms of CHM mapping between hyperbolic hairy AdS black hole and spherical R\'{e}nyi entropy in dual field theories. Based on these solutions, we study the temperature dependent condensation of dual operator of massive and massless scalar respectively. These condensations show that there might exist phase transitions in dual deformed CFTs. We also compare free energy between asymptotically hyperbolic AdS black hole solutions and hyperbolic AdS Schwarz (AdS-SW) black hole to test phase transitions. In order to confirm the existence of phase transitions, we turn on linear in-homogenous perturbation to test stability of these hyperbolic hairy AdS black holes. In this paper, we show how potential parameters affect the stability of hyperbolic black holes in several specific examples. For general values of potential parameters, it needs further studies to see how the transition happens. Finally, we comment on these instabilities associated with spherical R\'{e}nyi entropy in dual deformed CFTs.