Thermal contact resistance (TCR) is a critical parameter influencing the temperature distribution in high-temperature thermal structures, particularly aerospace applications. In optimization design, neglecting TCR may lead to structural failure or unnecessary redundancy. While topology optimization for mechanical contact problems has been extensively studied, research on thermal contact remains relatively limited. This work presents a topology optimization method for multilayer thermo-elastic structures based on a Galerkin finite element method which can address thermal contact problems. The minimum volume ratio is adopted as the optimization objective, while the total compliance and the maximum temperature of each layer are set as constraints to carry out light-weight design. The effectiveness of the method is demonstrated through two numerical examples, and the effects of TCR on the volume ratio and compliance ratio under different loading conditions are thoroughly analyzed. The volume ratio and compliance ratio of the design domain initially decrease and then increase with the rise in TCR. The introduction of an appropriate TCR not only enhances the stiffness of the design domain but also reduces the overall structural weight. In addition, the increase in mechanical load will improve the effect of TCR on light-weight design. This method provides valuable guidance for the lightweight design of thermo-elastic structures with TCR.