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Employing the instanton technique and the nonlinear perturbation method, we study quantum-classical escape rate crossover for several forms of magnetic anisotropy in a nanomagnet sandwiched between two superconductors called Josephson φ0 junction. It is shown that the phase boundary separating the first- and second-order crossover is greatly influenced by the bias current as well as the anisotropy constant and the external magnetic field. These features are expected to be observable with existing experimental techniques.