The pump beam for an end-pumped solid state laser can likely be fit by a super-Gaussian beam. We normalize the super-Gaussian beam for different orders (m). Numerical temperature solutions are obtained for super-Gaussian beams using the finite element method. We find that these numerical temperature solutions can be fit by a Gaussian function. This is important as all of the subsequent thermal radial and tangential stresses, refractive index changes, optical path differences (OPDs), birefringences and depolarization losses can be solved analytically and they follow Gaussian profiles. Different m have different thermal effects as well as different overlap efficiencies. We analyze the effects of different m on different pump powers, different pump beam sizes, and different rod radii. This analysis could provide valuable data for correcting phase distortions of high-power operation.