Abstract: Rayleigh-Taylor (RT) instability phenomenon exists widely in nature and engineering fields. It is of great theoretical significance and practical value to clearly understand the physical mechanism of the RT instability. In this paper, the compressible RT instability is simulated by the discrete Boltzmann method (DBM), and the compressible RT instability with random multimode initial perturbations at continuous interfaces is numerically investigated by means of the DBM. The results show that with the influence of temperature gradient, the thermodynamic non-equilibrium strength related to heat flux firstly increases and then decreases. Under the action of thermal diffusion, the thermodynamic non-equilibrium strength at the interface firstly decreases and then increases, which affects the time evolution of proportion of the thermodynamic non-equilibrium region. In this respect, effects of temperature gradient and thermal diffusion on the time evolution trend of non-equilibrium strength at the interface are the same. Finally, we analyze the time evolution of the global average thermodynamic non-equilibrium strength, and find that under the joint action of macroscopic physical gradients and thermodynamic non-equilibrium area, the global average thermodynamic non-equilibrium strength firstly increases, then decreases, and finally tends to be stable. On the one hand, the increase (decrease) of the area of thermodynamic non-equilibrium region will increase (decrease) the strength of thermodynamic non-equilibrium. On the other hand, the increase (decrease) of physical gradients at the material interface also has the same effect on the global average thermodynamic non-equilibrium strength. The two physical mechanisms interact and compete with each other.