We investigate the role of climate risk in the Eurozone sovereign debt market, to evaluate the current pricing of different risk factors in the government spreads of each country. Particular attention is paid to differences between green and non-green bonds, to assess the potential of sustainable debt to hedge climate risk. Weather variables are selected in line with the guidelines of Eurozone climate stress tests and taken as proxies of acute and chronic physical risk, while EU carbon allowances are included to capture transition risk. Their significance is studied as potential drivers of mean spread variations and for their co-movement with spreads in the case of extreme events, to provide insights for climate risk management and financial policy, with the goal of enhancing the resilience and stability of the financial system. From both analyses, the same results emerge: climate risk is being priced by the market, but differently depending on the country, and green government bonds from different countries have a divergent reaction to climate risk factors. Dutch and French green spreads closely mirror their traditional counterparts, while German, Italian, and Spanish green bonds exhibit greater hedging potential. Various explanations are considered, including a "scarcity effect" linking the behavior of green bonds to their abundance relative to total outstanding government debt. Finally, the most relevant risk factors of each country are highlighted, comparing the results to known climate change challenges.