Abstract: With the rapid development of hypersonic flight vehicles in the upper atmosphere, micro-electro-mechanical systems (MEMS), and extreme ultraviolet (EUV) lithography machines, there has been a growing focus on the analysis of flow stability under the combined effects of rarefied gas dynamics and thermal convection. However, conventional flow stability analysis methods based on the Navier-Stokes equations are unable to account for phenomena occurring in highly rarefied flows. Furthermore, the Boltzmann-BGK (Bhatnagar-Gross-Krook) model equation assumes a constant Prandtl number of 1, which does not accurately represent real physical situations, particularly when dealing with varying thermodynamic properties. To address these limitations, a new linear stability analysis method has been developed starting from the Boltzmann-Shakhov model equation, which allows for an adjustable Prandtl number across different flow regimes. This method is used to study the stability of Rayleigh-Bénard convection under varying levels of rarefaction. The results show that Rayleigh-Bénard flow is highly susceptible to instability when the density distribution between the upper and lower walls transitions from an "upper-light, lower-heavy" state to a "lower-light, upper-heavy" state. This transition is particularly critical for flow stability. Additionally, the growth rate exhibits a characteristic turning point at a specific wavenumber, which could be explained by the change in the number of extrema in the perturbation eigenfunction from one to two. As the degree of rarefaction increases, the Prandtl number corresponding to the most unstable mode decreases gradually, indicating the influence of rarefaction on the flow's stability behavior. This study provides deeper insight into the complex interaction between rarefied gas effects and thermal convection in flow stability.