The formation of a cusp on a black hole shadow is a striking signature of physics beyond the Kerr paradigm. Using the Konoplya-Zhidenko metric, a general parametrized deformation of the Kerr black hole, 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.

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• Classical black holes
• Critical phenomena
• General relativity formalism
• Quantum aspects of black holes