  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.
