We numerically demonstrate optical bistability in a nonlinear multilayer structure by utilizing the unique dispersion of hyperbolic metamaterials. The linear transmission is varied sharply with topological transition of isofrequency contour of the multilayer structure, and this non-resonant scheme enables realization of optical bistability with a short response time and a relatively low switching intensity. We have investigated exhaustively all possible topological transitions in the dispersion characteristics of the multilayer structure for optical bistability, and shown that the hyperbolic metamaterial (HMM) type transition from Type II to Type I, and the transition from Type II HMM to effective dielectric are suitable for realizing high-performances optical bistability. The proposed schemes can overcome the trade-off between a switching intensity and a response time in resonance based optical bistabilities.