We study the impact of climate scenarios on sovereign credit risk, measured by Credit Default Swaps (CDS), analysing the interplay of climate risk exposure, fiscal and financial characteristics of European countries. We develop and estimate a panel threshold model using annual data from 2010 to 2022 for 24 European countries. The country's indebtedness level defines the threshold of the estimated regimes. Then, we project the path of sovereign risk up to 2050 using the climate scenarios of the Network for Greening the Financial System. Results show a temporal and structural dimension of climate sovereign risks. First, in the short term, sovereign risk may worsen due to GDP and fiscal revenue adjustments, and for some countries, particularly in a delayed transition. However, in the long run, an orderly (Net Zero 2050) transition brings co-benefits in terms of lower sovereign risk, compared to a delayed transition scenario. Second, countries with a debt-to-GDP ratio equal to or higher than 60% are the most sensitive to increased credit risk conditioned to scenarios of delayed transition and chronic physical risk, due to the economic and financial implications induced by stranded assets. Third, climate sovereign risk is stronger in economies based on traditional sectors, and in countries with higher political instability. Our results show the importance of European governments credibly committing to an orderly and early Net Zero transition to preserve sovereign financial stability.