<p>This paper numerically studies if the stability of a stable black hole is robust against the small perturbation on geometry near its event horizon. In other words, we numerically study if two nearly identical black holes may exhibit completely different stabilities at late time. As a toy model, it encodes such perturbation into deformations of Regge-Wheeler potential. It considers three different types of local deformations—the negative static bump potential, the stochastic potential and bump potential modulated by time function in the low-frequency limit. Our numerical results show that infinitesimal local deformations on Regge-Wheeler potential near the horizon can overturn stability of a stable black hole, implying that late-time behavior of a stable black hole is extremely sensitive to geometry near horizon. Specifically, certain deformations that stabilize systems in flat backgrounds can destabilize otherwise stable black holes. It also shows that horizon-induced redshift transforms near-horizon quantum fluctuations into classical-scale stochastic deformations capable of triggering instability, implying that even an isolated black hole cannot remain stable if the near-horizon quantum noise could be held on extended timescales.</p>