High entropy alloys is an emerging class of materials with superior mechanical properties down to cryogenic temperatures. At 77K, and unlike traditional metallic alloys, an increase in strength, strain hardening rate, and ductility has been reported. This enhancement in properties has been attributed to the activation of twinning as an additional deformation mechanism at low deformation temperatures. The tendency for the formation of twinning and the hardening response dependence on crystal orientation has not been fully explored. This study is dedicated to explore the deformation evolution across several crystal orientations for the equiatomic FeNiCoCrMn high entropy alloy at room temperature (RT) and 77K. The works aims to establish the critical resolved shear stresses (CRSS) for slip and twinning and study the orientation and temperature dependence in these magnitudes. The experimental results have revealed a strong temperature dependence in the CRSS for slip, increasing from 56 MPa at RT to 155 MPa at 77K, with negligible orientation dependence. At 77K, not all crystal orientations developed twinning even at high levels of deformation. The lack of twinning has been attributed to differences in the hardening response resulting in low stress levels below the twinning CRSS of 153 MPa, as established in this work. No twinning was observed in any of the crystal orientations deformed at room temperature, regardless of the level of hardening and the achieved stresses. Overall, the results discussed in this work enhances our understanding of the local deformation response in single crystalline FeNiCoCrMn high entropy alloy, particularly the nucleation of slip, nucleation of twinning, and the effect of crystal orientation and loading temperature on these deformation mechanisms.