The Institute of High Energy Physics of the Chinese Academy of Sciences completed the research and development of the high quality factor 1.3 GHz superconducting cryomodule in June 2023, taking the lead in the world to realize the technical route of the medium temperature baking. Eight 1.3 GHz 9-cell superconducting cavities with the medium temperature baking process are integrated. During the integration test of the cryomodule, the temperature of the high-order mode (HOM) coupler of the superconducting cavity was abnormal, which made the superconducting cavity unable to work stably under high gradient. In this paper, the electromagnetic analysis of the high-order-mode coupler is carried out by the HFSS software and eigenmode Solver in CST software and the thermal analysis of the high-order-mode coupler is carried out by theory and Ansys Workbench software. Combining with the high-power experiment of cavity, the cause of the abnormal performance of the superconducting cavity was found. Also, the cooling structure of the HOM coupler in the superconducting cavity was further optimized to solve the instability of the superconducting cavity under high gradient in the module.