Gravitational equal-area law and critical phenomena of cuspy black hole shadow

The formation of a cusp on a black hole shadow is a striking signature of physics beyond the Kerr paradigm. We demonstrate that this morphological change fundamentally alters the shadow's topology with the topological charge flipping from 1 to -1. To analyze this topological transition, we introduce a gravitational equal-area law, analogous to Maxwell's construction in thermodynamics, and identify a critical point for cusp formation. Near this point, we uncover universal behavior characterized by a critical exponent 1/2, which places this gravitational lensing system within the mean-field universality class. These results establish a new framework for testing fundamental physics of black hole shadows, reframing the search for deviations from general relativity as a targeted hunt for a distinct topological and critical phenomenon.