For a broad class of hybrid dynamical systems, we establish results for robust asymptotic stability of hybrid limit cycles with multiple jumps per period. Hybrid systems are given in terms of differential and difference equations with set constraints. Hybrid limit cycles are given by compact sets defined by periodic solutions that flow and jump. Under mild assumptions, we show that asymptotic stability of such hybrid limit cycles is not only equivalent to asymptotic stability of a fixed point of the associated Poincare map but also robust to perturbations. Specifically, robustness to generic perturbations, which capture state noise and unmodeled dynamics, and to inflations of the flow and jump sets are established in terms of KL bounds. A two-gene network with binary hysteresis is presented to illustrate the notions and results throughout the paper.