Using meteorological monitoring data from eight climate disasters across the globe, the Climate Policy Uncertainty (CPU) Index, and the financial data of eight energy markets between 2010 and 2020, this study employs the fixed base extreme difference entropy method to construct a Climate Risk Index (CRI) and investigates the impact of climate risk on the risk spillover among energy markets based on dynamic heterogeneous network analysis. The results confirm that the CRI fluctuated periodically and exhibited an upward trend during the sample period. In terms of risk spillover, the oil, geothermal, and fuel cell markets act as risk receivers, whereas the gas, wind, and bioenergy markets functions as risk emitters. Climate risk significantly increases the network complexity between energy markets and the risk spillover of gas, solar, water, and wind markets but decreases the risk spillover of oil, bioenergy, fuel cells, and geothermal markets. Additionally, the fossil fuel market is more sensitive to climate risk than is the renewable energy market.