Optical and thermodynamic properties of Kerr-Bertotti-Robinson black holes

We investigate the thermodynamic and optical properties of Kerr–Bertotti–Robinson black holes, namely rotating black holes immersed in an external Bertotti–Robinson electromagnetic background. In the fixed-$a$ ensemble, we derive the horizon mass relation, the Hawking temperature, the entropy, the Helmholtz-type free energy, the heat capacity, and the extremal remnant configuration. These quantities reduce smoothly to their Kerr counterparts as $B\to 0$. In the weak-field regime, the leading thermodynamic corrections arise at order $B^2$; the extremal radius is shifted at this order, whereas the remnant mass receives its first correction only at order $B^4$. We also introduce a formal AdS-like thermodynamic interpretation of the Bertotti–Robinson scale, treating the associated pressure as an effective response variable rather than a genuine cosmological pressure. Because the spacetime is not asymptotically flat, we further compute the finite-radius Komar mass and the Komar charge associated with the horizon generator. Using the Hamilton–Jacobi formalism, we derive the separated null-geodesic potentials, the impact parameters of spherical photon orbits, and the celestial coordinates of the shadow boundary for a finite-distance observer. We then characterize the photon-region boundaries, ergosphere thickness, photon–ergosphere gap, shadow area, and magnetic shadow susceptibility. Within the perturbative regime considered, the Bertotti–Robinson background decreases the averaged ergosphere thickness and shadow area, increases the photon–ergosphere gap, and produces a negative shadow susceptibility whose magnitude is enhanced by rotation.