This study investigates the time- and frequency-domain spillover dynamics between climate-risk indices, namely the Transition Risk Index (TRI), Physical Risk Index (PRI), Global Climate Policy Uncertainty (GCPU))and major energy futures markets, including ICE Europe Brent crude oil futures (continuation), the global crude benchmark (Brent), ICE Europe Low Sulphur Gasoil futures, a European middle-distillate benchmark (Gasoil), Intercontinental Exchange (ICE) Abu Dhabi Murban crude oil futures (continuation), a Middle Eastern light-sweet benchmark (Murban), Shanghai Crude, New York Harbor Ultra-Low Sulphur Diesel futures (NYMEX) continuation (ULSD), and West Texas Intermediate crude oil futures (NYMEX Light Sweet Crude Oil futures continuation) (WTI). Employing a flexible econometric framework based on TVP-VAR and quantile connectedness, the analysis uncovers non-linear, asymmetric, and time-varying spillovers, with markedly stronger linkages during extreme market conditions and in the short term. Energy commodities, particularly Gasoil and WTI, emerge as significant net transmitters of transition risks, amplifying volatility during periods of stress, while long-term spillovers remain relatively weak, reflecting gradual decarbonization trends. The unique contribution of this paper lies in extending the Arbitrage Pricing Theory (APT) by integrating climate risks as dynamic, state-dependent, and non-diversifiable factors, thereby demonstrating how energy asset sensitivities fluctuate across regimes and quantiles. This approach advances the asset pricing and climate finance literature beyond static models by embedding dynamic connectedness into risk transmission analysis. The findings highlight the systemic nature of climate risks and underscore the importance of adaptive financial regulation, forward-looking climate policy, and flexible risk management practices to mitigate volatility and support the global energy transition. These insights provide actionable guidance for policymakers, regulators, and investors navigating the evolving interplay between climate risks and energy markets.