Carbon-intensive firms face mounting pressure to develop substantive corporate climate risk management (CCRM), yet its firm-level and country-level antecedents remain unevenly understood. Drawing on stakeholder and institutional theory, we examine three drivers of CCRM: sustainability governance, voluntary climate-membership commitments, and regulatory quality. Our data cover 1,295 firm-year observations across 43 countries over 2018–2022. We estimate ordered logistic regressions with lagged regressors, with ordered probit, two-step system GMM, and sub-sample robustness checks. In the main specification, sustainability governance and regulatory quality are both positive antecedents (β = 2.441 and β = 1.676, p < 0.001); climate membership exerts a sector-conditional effect concentrated in energy and basic materials. Sub-sample analyses reveal that internal governance dominates among non-state-owned firms, while among state-owned firms (a sub-sample heavily concentrated in Chinese SOEs) regulatory quality dominates instead. We frame the latter as suggestive context-conditional substitution rather than a universal feature of state ownership. CCRM is highly persistent (system-GMM lagged coefficient = 0.693, p < 0.001), suggesting that climate risk management is best understood as a path-dependent organizational capability built incrementally over time. Firms strengthening CCRM should invest in integrated governance architecture; regulators should treat regulatory-quality reform as complementary to direct climate mandates.