We obtained analytic solutions of boundary conditions to the stable region of Z-fold $Cr^{4+}$ : YAG laser cavity when the conditions are with and without thermal tensing effect. Also we investigated the influence of the thermal tensing effect on the stability of cavity, beam waist, and astigmatic compensation using aberration transformation matrices. The thermal tensing effect almost has no influence on the stable region of the cavity when the crystal is located in the middle of two concave mirrors and when the distances from the concave mirror to the reflecting mirror and the output coupler are the same. The beam waist, however, is affected more in a tangential plane than in a sagittal plane, and so it is difficult to have astigmatic compensation when the thermal tensing effect exists. This result means that the thermal tensing effect should be considered even for the Kerr-lens mode-locking.