High energy laser beams propagating in the atmosphere are subjected to a variety of effects, such as the absorption and scattering of molecule and aerosol, atmospheric turbulence effects, thermal blooming effects, and the interaction between turbulence and thermal blooming. In general, these atmospheric propagation effects degrade laser beam quality and reduce the beam power concentration at the target. With adaptive optics compensation, the beam quality can be modified. But small-scale perturbation has developed and the phase compensation becomes unstable in some conditions. The performance of adaptive-optics system is degraded, which effects can be well explained by small-scale linear theory of thermal blooming. However previous theoretical studies of small-scale thermal blooming focused on the Kolmogorov turbulence. In the past decade, experimental evidence has shown significant deviations from Kolmogorov model in certain portions of the atmosphere. An generalized power-law of non-Kolmogorov turbulence model has been introduced, which becomes quite popular in the optical propagation community. Numerous theoretical and developmental efforts have been made based on non-Kolmogorov turbulence model in recent years. Thus it is very meaningful and imperative to explore the theoretical mechanism of high energy laser phase compensation with non-Kolmogorov turbulence.In this study, the Strehl ratio of the thermal blooming phase compensation is generalized with the non-Kolmogorov turbulence spectrum, and the analytical expression is obtained based on the linear theory of small-scale thermal blooming. The influence of the turbulence spectrum on the phase compensation of the high energy laser is analyzed. The results show that the turbulence spectrum has an important influence on the phase compensation of turbulent thermal blooming effect. Under the same turbulence Fresnel number condition, the compensation effect is worse when the spectral index is closer to 3 and the compensation effect is better when the spectral index is close to 4. Under the same atmospheric coherence length condition or under the same turbulence refractive index constant condition, the Strehl ratio decreases with the increase of the thermal blooming effect when the spectral index is close to 3 and the decline rate of the Strehl ratio is slower when the turbulence spectrum index is close to 4. This is because as the turbulence spectrum exponent increases, the logarithmic amplitude fluctuation slows down due to the interaction between turbulence and thermal blooming. These theoretical results can provide some scientific bases and theoretical guidance for the practical applications of high energy laser transmission.