The three-dimensional heat deposition of the cryogenic Yb:YAG regenerative amplifier, which is yielded by pulsed pumped laser, is derived in details based on the theory of quasi-three-level rate equations. Furthermore, the transient temperature field, stress, and strain induced by the thermal gradients in the laser crystal are analyzed by use of the finite element method. Then the thermally induced lens and depolarization in the cryogenic regenerative amplifier are theoretically studied. We find that for the pump and cooling structure which has been designed, the focal length of the thermally induced lens is about 15m and the depolarization rate could be ignored. The maximum output energy 10.2 mJ at a repetition rate of 10 Hz with nearly TEM00 mode profile is obtained using the designed pump and cooling structure.