We study the effect of the Hawking-Page phase transition on the Page Curve
for the BTZ black hole coupled to two thermal baths and with the inclusion of
island. The Page Curve is shown by us when the Hawking-Page phase transition
happens. The results show that the Hawking-Page phase transition may happen
before the Page time or after the Page time under certain conditions. We also
study the holographic entanglement entropy when the Hawking-Page phase
transition happens for the BTZ black hole in the gauge/gravity setup and derive
a first law of entanglement in the boundary and bulk. If a first law of
entanglement thermodynamics is universally true, then we can conjecture that
all the information of the black hole will come out at the critical temperature
when the Hawking-Page phase transition happens. Furthermore, We further
speculate that the effect of the first-order phase transition of black holes on
the black hole information is universal. These results mean that black holes
information does not have to escape by the radiation, and the first-order phase
transition of black holes can also cause information to escape.
