Gravity/thermodynamics correspondence via black hole shadows

The shadow of a black hole serves as a pristine window into the strong-gravity regime, with cuspy feature emerging as a smoking-gun signature of physics beyond the Kerr paradigm. In this paper, we extend the work of [arXiv:2601.15612 [gr-qc]] and study the detailed properties of the cuspy shadow by using the parametric expressions of the shadow boundary. From a topological perspective, we provide a rigorous topological classification of these shadows, categorizing them into distinct ``rectangular"and ``8-shape"topologies. Crucially, we establish a formal gravity/thermodynamics correspondence by mapping the cuspy shadow to the swallowtail behavior observed in thermodynamic free energy. We demonstrate that the self-intersection of the shadow boundary, marking a geometric phase transition, can be precisely determined through three independent but equivalently thermodynamic-like approaches. Furthermore, we analytically derive the critical exponents governing the emergence of these cusps, revealing that they are consistent with the mean-field universality class. Our results suggest that the observational features of black hole shadows are deeply rooted in the underlying gravitational thermodynamics, offering a novel framework to probe the fundamental nature of spacetime.