Abstract: High damping rubber material has been widely employed in seismic design of buildings and bridges, primarily due to its exceptional energy dissipation capabilities. This material effectively absorbs and dissipates seismic energy, significantly reducing the potential for structural damage. Previous studies have demonstrated that the mechanical behavior of high damping rubber is primarily influenced by temperature, loading rate, experienced amplitude, and other factors. The majority of existing high damping rubber constitutive models have focused on one or two of these factors, and have been unable to consider the influence of the Mullins effect, which has led to some limitations. In light of the aforementioned considerations, a multi-case variable amplitude cyclic compression-shear test of high damping rubber specimens was conducted with the objective of studying the effects of temperature, loading rate, and experienced amplitude on the mechanical behavior of high damping rubber. The high damping rubber constitutive model was decomposed into the superposition of a hyperelastic part and a viscoelastic part. A time-temperature equivalent equation considering the temperature effect and a damage function considering the Mullins effect are introduced. Finally, a high damping rubber thermal-hyper-viscoelastic constitutive model with Mullins effect is obtained. The optimize module in the third-party library Scipy of Python is then used to fit the test data step by step by nonlinear least square method, and the corresponding constitutive model parameters are obtained. Finally, the variation of parameters is employed to fit the equations of parameters varying with temperature and strain rate. Some experiments are conducted to verify the validity and applicability of the proposed high damping rubber thermal-hyper-viscoelastic constitutive model. The results demonstrate that the proposed high damping rubber thermal-hyper-viscoelastic constitutive model can accurately describe the complex stress-strain relationship of high damping rubber materials, and it can also describe the effects of temperature, loading rate, and experienced amplitude. Furthermore, the model has good applicability.