  We study the growth of entanglement entropy in a doubly holographic model of
gravity for a spherical AdS black hole. Compared to previous work, which was
limited to the case of planar black holes, this introduces an extra scale to
the problem. This allows us to analyze the interplay between the reorganization
of entanglement entropy due to island formation and the onset of the
Hawking-Page phase transition and to find the appearance of a new critical
black hole radius unrelated to the thermodynamics. We also find that the
geometry of the Ryu-Takayanagi surface capturing the physics of islands
exhibits drastically different behavior than in the planar case.
