Deciphering black hole phase transitions through photon spheres

Black hole thermodynamics is a crucial and foundational aspect of black hole physics, yet its observational verification remains exceptionally challenging. The photon sphere of a black hole, a manifestation of strong gravitational effects, is intrinsically linked to its shadow, which has been directly captured through observations made by the Event Horizon Telescope. Investigating black hole thermodynamics from a gravitational perspective presents an intriguing avenue for research. This paper obtains an analytical formula for the coexistence curve and investigates the relationship between the thermodynamic phase transition and the photon sphere of a black hole with quantum anomaly. It proposes that the photon sphere encodes information about the black hole phase transition, arguing that the change in the photon sphere radius can serve as an order parameter characterizing the black hole's phase transition.