In this study, the detailed harmonic analysis of GaN high electron mobility transistor (HEMT) at different temperatures and frequencies is presented. Volterra power series and multi-dimensional Laplace transform are used as a method. The Volterra power series is also solved up to third degree, and the small signal transfer functions of kernels (H-1, H-2 and H-3) are obtained. The relationship between drain inductance (L-d), gate-source voltage (V-gs), impedance (Z(L)) and the effect of frequency (F-r) to the output gain is identified. Besides, the nonlinear gains of H-1, H-2 and H-3 kernels of the GaN-HEMT are obtained. The inverse relationship between the output gains of H-1, H-2 and H-3 kernels are derived. An unsuitable situation has also been identified for sub-carrier inter-modulation systems. In addition, an asymmetric structure is also obtained between the output gain of H-2 and side-band frequencies. The effects of other parameters are carried out for the output gain.