A remaining carbon budget (RCB) estimates how much CO2 we can emit and still reach a specific temperature target. The RCB concept is attractive since it easily communicates to the public and policymakers, but RCBs are also subject to uncertainties. The expected warming levels for a given carbon budget has a wide uncertainty range, which we show here to increase with less ambitious targets, i.e., with higher CO2 emissions and temperatures. Leading causes of RCB uncertainty are the future non-CO2 emissions, Earth system feedbacks, and the spread in the climate sensitivity among climate models. The latter is investigated in this paper, using simple emulators of Earth System Models in the Coupled Model Intercomparison Project Phase 6 (CMIP6) ensemble. It is shown that the transient climate response to cumulative emissions of carbon (TCRE) is approximately proportional to the effective equilibrium climate sensitivity (ECS). For temperature targets between 1.5-3.0 degrees C, the models exhibiting low ECS increase RCB by a factor two compared to those with high sensitivity, suggesting that observational constraints imposed on the ECS in the model ensemble also will reduce uncertainty in the RCB estimates.