This paper presents a physics-based model for computing the combined effect of applied strain and temperature on the device characteristics of aluminium gallium nitride (AlGaN/GaN) high electron mobility transistors (HEMTs). More specifically, the electrical response of the HEMT is predicted under applied biaxial strain from ±1% over a wide range of temperatures (300-500 K). In addition, the interface state densities at the Schottky-AlGaN interface are introduced in the model. This physics-based model calculates the charge due to applied, thermal and lattice mismatch strain and temperature effects at the two-dimensional electron gas (2DEG) interface of the HEMT. Coupled with a model for the 2DEG mobility that includes strain and temperature effects, current-voltage characteristics for the HEMT are derived above the threshold voltage. Regimes with large strain sensitivity and temperature compensation are identified and vice-versa. The analysis from the model clarifies the large range of strain response variations observed in the experimentally measured characteristics of HEMTs in literature. Furthermore, the developed model is a useful tool for predicting the response of HEMTs used in sensing and under the influence of packaging in extreme environments, especially when temperature fluctuation and strain coupling is of concern.