Climate feedbacks generally become smaller in magnitude over time under CO2 forcing in coupled climate models, leading to an increase in the effective climate sensitivity, the estimated global-mean surface warming in steady state for doubled CO2. Here, we show that the evolution of climate feedbacks in models is consistent with the effect of a change in tropospheric stability, as has recently been hypothesized, and the latter is itself driven by the evolution of the pattern of sea-surface temperature response. The change in climate feedback is mainly associated with a decrease in marine tropical low cloud (a more positive shortwave cloud feedback) and with a less negative lapse-rate feedback, as expected from a decrease in stability. Smaller changes in surface albedo and humidity feedbacks also contribute to the overall change in feedback, but are unexplained by stability. The spatial pattern of feedback changes closely matches the pattern of stability changes, with the largest increase in feedback occurring in the tropical East Pacific. Relationships qualitatively similar to those in the models among sea-surface temperature pattern, stability, and radiative budget are also found in observations on interannual time scales. Our results suggest that constraining the future evolution of sea-surface temperature patterns and tropospheric stability will be necessary for constraining climate sensitivity.