[en] Phase-field damage models are able to describe crack nucleation as well as crack propagation and coalescence without additional technicalities, because cracks are treated in a continuous, spatially finite manner. Previously, we have developed a phase-field model to capture the rate-dependent failure of rubber, and we have further enhanced it to describe failure due to cyclic loading. Although the model accurately describes fatigue failure, the associated cyclic simulations are slow. Therefore, this contribution presents an acceleration scheme for cyclic simulations of our previously introduced phase-field damage model so that the simulation speed is increased to facilitate large-scale simulations of industrially relevant problems. We formulate an explicit and an implicit cycle jump method, which, depending on the selected jump size, reduce the calculation time up to 99.5%. To circumvent the manual tuning of the jump size, we also present an adaptive jump size selection procedure. Thanks to the implicit adaptive scheme, all material parameters are identified from experiments, which include fatigue crack nucleation and crack growth. Finally, the model and its parameters are validated with additional measurements of the fatigue crack growth rate.