Combining the split-step Fourier method with the fourth-order Runge-Kutta integration method, the manipulation and generation of supercontinuum by finite energy cosh-Airy pulses in a double-zero dispersion media was investigated. Firstly, the influences of truncation coefficient a, initial chirp C and distribution factor χ₀ on the evolution of cosh-Airy pulses in double-zero dispersion medium were discussed in detail, and the influences of a, C and χ₀ on the width of supercontinuum width were statistically analyzed. Then, the influences of higher-order nonlinear effects on supercontinuum generation of cosh-Airy pulse was further studied. The results show that the width of supercontinuum can be controlled by manipulating the characteristic parameters of cosh-Airy pulses. The flatness of supercontinuum is affected with consideration of high-order nonlinear effects. The results provide some theoretical basis for manipulation and generation of supercontinuum and broadband laser sources.