Optical bistable responses in AlGaAs-based photonic crystal cavities with Al fractions of 0.16 and 0.27 under a stationary condition are studied. An analysis of the experimental results with a one-dimensional etalon model indicates that the bistable response is primarily driven by optical nonlinearities that stem from a heat generation in two-photon absorption and in free-carrier absorption. A free-carrier plasma effect is not influential, which is a major difference from the case of Si photonic crystal cavities. The accumulation of heat enables the phase or index to significantly change. An estimate indicates that a high $Q$-factor is advantageous for driving bistability exclusively by the Kerr effect.