Empirical estimates of equilibrium climate sensitivity (ECS) can be obtained by comparing measurements of short-term radiation changes at the top of the atmosphere during the satellite era to corresponding changes in surface temperatures. Most estimates of ECS utilized a global energy balance model where all quantities are global and annual averages.
Ray Bates estimates ECS using a two zone energy balance model, where the radiative responses in the tropics (30 N to 30 S) and extratropics are estimated separately, and the dynamic heat transport from the tropics to the extratropics are explicitly estimated that depends linearly on the difference between the tropical and extratropical temperature perturbations. He used observations of the radiative response as reported by Lindzen and Choi  and Mauritsen and Stevens . The radiative response is the change in the net upward longwave plus shortwave flux at the top of the atmosphere per change in the surface temperature. He chose likely ranges of the three parameters and finds that the calculated ECS is tightly constrained, with a likely range of 0.85 °C to 1.28 °C. The best estimate is 1.02 °C, which is the median of estimates calculated from the midpoints and ends of each range of the input parameters.
This estimate implies that continued exponention increase in the CO2 concentration in the atmosphere will cause a temperature increase of about 0.57 °C from now to 2100. See the Ray Bates paper here.
The diagram below shows several recent estimates of ECS in the scientific literature.