We explore theoretically the optomechanical interaction between a light field and a mechanical mode mediated by a Kerr nonlinear medium inside a Fabry-Perot cavity. When the system is driven by a strong and fast amplitude-modulated light field, i.e., in the so-called temporal rocking region, the cavity field and the mechanical oscillator show the characteristics of multistability. The rocking breaks down the continuous phase symmetry of the cavity field to a bistable case of two equivalent states with exact π phase difference. In addition, the rocking can significantly enhance the optomechanical coupling between the light field and the mechanical oscillator, which can be used as a new handle to control the normal mode splitting of the mechanical spectrum. Moreover, the optomechanical cooling rate can be greatly modified by the rocking. With the optimized rocking parameters, the mechanical oscillator can be cooled down to its ground state more efficiently. Such a temporal rocking optomechanical system has potential applications in all-optical switching and enhancement of quantum effects.