Climate change is the most important challenge currently faced by humans, which affects their health and financial stability. This study proposes a multilayer network that includes return, volatility, and risk layers to investigate the risk connectedness between climate policy uncertainty and the sectoral stock market in China. We explore the static and dynamic topological features of multilayer networks from the perspectives of the system and market levels, respectively. The results indicate that (i) a crisis increases the intensity of risk spillovers; (ii) an obvious nonsynchronous effect was found among the layers during financial turmoil; and (iii) industrials, materials, information technology, and consumer discretionary act more often as net risk transmitters during the sample period, whereas climate policy uncertainty, telecommunication services, and financials are more frequently net risk receivers. The results of this study have important implications for regulators, policymakers, and investors in alleviating climate risk and developing hedging and investment strategies.