While the potential of high entropy alloys (HEAs) as high strength structural materials has been recognized, the specific deformation mechanisms for each alloy system have not yet been fully identified or described conclusively. Especially, their mechanical properties depend significantly on the microstructural details and the interaction of defects with the different microstructural components. Here, to elucidate the influence of the crystal structure on the deformation mechanisms of HEAs in the temperature range from 293K to 573K, high-temperature nanoindentation experiments have been performed on two different HEA systems, i.e. CoCrFeMnNi and NbMoCrTiAl, which are representative single-phase solid-solution strengthening alloys with face-centered cubic (fcc) and body-centered cubic (bcc) structure, respectively. In general, contrary to fcc metals where the plastic deformation exhibits a pronounced rate dependence at elevated temperature, the rate-sensitive deformation in bcc metal is observed at room temperature and disappears at high temperature. Compared to common metals, both HEAs show different temperature- and rate-dependent behaviors. These phenomena may be related to different kinetics of thermally activated dislocation motion, which governs the slip and the plastic resistance. Therefore, the observed signatures from nanoindentation experiments, e.g., strain-rate sensitivity, activation volume and activation energy, are discussed in the context of the active slip systems and the thermally activated motion of dislocations for the different crystal structures.