Central banks now intervene in climate governance through climate-related communication, but we know little about how this affects financial markets. We examine whether such discourse is associated with differences in how long-term sovereign debt prices reflect climate risk. Building on salience theory, we develop a framework in which climate communication increases yields when it makes climate-related risks more salient and decreases yields when it reduces policy uncertainty. Institutional authority and credibility amplify these effects. Using a dataset of over 17,000 speeches from 107 central banks between 2000 and 2021, we construct an index of climate related talk and estimate two-way fixed effects models for 10-year sovereign yields. We find that sovereign yields are higher when central banks make climate risks more salient, consistent with a climate risk premium. The market response is stronger in countries with more independent central banks, suggesting that institutional authority shapes how investors interpret governance signals. The results imply that central bank communication is not merely symbolic but part of the institutional infrastructure through which climate risk is priced, with implications for states’ climate policy capacity and for the distribution of climate transition costs.