  This study focuses on the examination of the island rule within the context
of four-dimensional Reissner-Nordstr\"om-AdS (4D RN-AdS) black holes,
illuminating the intricate relationship between the entanglement entropy and
phase transitions of black holes. The entanglement entropy of 4D RN-AdS black
holes follows the anticipated linear growth pattern before ultimately declining
to a constant value, in accordance with the well-established Page curve. The
novelty of this study lies in the examination of the influence, previously
unexplored, of the first-order phase transition on the shape and evolution of
the Page curve in situations involving both eternal and evaporating black
holes. Despite the morphological alterations of the curve induced by the
transition, the inherent unitarity of the system persists. As the evaporation
progresses, the Page curve displays diverse configurations, unveiling phenomena
that are novel and defies traditional expectations, thereby enriching our
comprehension of the thermodynamics of black holes interlinked with quantum
information.
