Photon orbits and phase transition for gravitational decoupled Kerr anti-de Sitter black holes

Interpreting the cosmological constant as the energy of the vacuum and using a gravitational decoupling approach leads to a new Kerr–anti-de Sitter (AdS) black hole. The metric of the new Kerr–AdS is more straightforward than the standard Kerr–AdS and geometrically richer, showing the rotation’s impact as a warped curvature. We investigate the relationship between the unstable photon orbits and thermodynamic phase transition to the new Kerr–AdS black hole background. We derive an exact expression for thermodynamic properties of black holes, including mass ($M$), Hawking temperature ($T$), entropy ($S$), heat capacity ($G$), and free energy ($G$), by relating the negative cosmological constant to positive pressure through the equation $P=-\Lambda /8\pi =3/8\pi l^2$, where $l$ represents the horizon radius, and by introducing its conjugate variable as the thermodynamic volume $V$. When $P<P_c$, black holes with $C_P>0$ exhibit stability against thermal fluctuations, while those with $C_P\le 0$ are unstable. Our analysis of Gibbs free energy reveals a phase transition from small globally unstable black holes to large globally stable ones. Additionally, investigating the system’s $P-V$ criticality and determining the critical exponents shows that our system shares similarities with a Van der Waals (vdW) fluid. In the reduced parameter space, we observe nonmonotonic behaviours of the photon sphere radius and the critical impact parameter when the pressure is below its critical value. It indicates that alterations in the photon sphere radius and the minimum impact parameter can act as order parameters for the phase transition between small and large black holes. In discussing the applicability of the Maxwell equal area law, we highlight the presence of a characteristic vdW-like oscillation in the $P-V$ diagram. This oscillation, denoting the phase transition from a small black hole to a large one, can be substituted by an isobar. Furthermore, we present the distribution of critical points in parameter space and derive a fitting formula for the co-existence curve.