  Considering a doubly holographic model, we study the black hole information
paradox for the eternal AdSd-RN black hole coupled to and in equilibrium with a
d-dimensional conformal bath whose state has been deformed by the charged
scalar field coupled to a U(1) gauge field. Without a brane, the spontaneous
symmetry breaking of the gauge field on boundary systems can induce a
second-order phase transition of the charged scalar field at the critical
temperature, known as holographic superconductors. The bath deformation can
significantly change its entanglement dynamics with the black hole, resulting
in variations in the Page curve and Page time. Our results indicate that
characteristic parameters of the Page curve, such as entanglement velocity,
initial area difference and Page time, can be used as suitable probes to detect
superconducting phase transitions. In particular, the entanglement velocity can
also probe both Kasner flows and Josephson oscillations. When keeping the
endpoint of the radiation region fixed at twice the critical Page point, the
entanglement velocity (the internal backreaction) has a more significant
influence on the Page time compared to the initial area difference (the
external backreaction).
