This article investigates robust design optimization of turbofan engines to minimize climate impact while accounting for significant uncertainties in climate models based on performance calculations. By incorporating engine performance, flight mission, emission, and linear climate models, this work evaluates the impact of conventional kerosene-based aircraft engine emissions, including CO2, NOx, and contrails, on the climate, quantified using the average temperature response. The methodology compares deterministic and robust optimization approaches, including efficient multifidelity methods based on Cokriging surrogate models, to evaluate the robustness of climate impact reductions in new engine designs. Uncertainty quantification methods are applied to develop high- and low-fidelity models, enabling robust evaluations of climate impact reductions. The baseline design of a turbofan engine typical for an A320-class aircraft is analyzed to assess climate impact, robustness, and sensitivities to performance, emission, and climate model parameters. The results demonstrate that robust optimization, targeting the 95% quantile of the average temperature response, yields designs with reduced overall climate impact and lower uncertainty compared to deterministic optimization and the baseline. Specifically, changes in cruise altitude and engine pressure ratios reduce the 95% quantile by up to 41%. Furthermore, the findings highlight the importance of addressing NOx and contrail uncertainties, which contribute the most to variability in climate impact estimates. This work presents a novel framework that demonstrates the feasibility of integrating robust climate impact optimization in engine design while mitigating the risks posed by climate model uncertainties, marking a significant step toward more sustainable aviation. Extending its application to emerging propulsion concepts, such as sustainable aviation fuels or hydrogen-based engines, represents a promising direction for future research.