Some studies suggest that specific equilibrium climate sensitivity S might be state-dependent. Reanalyzing existing paleodata of global mean surface temperature ∆Tg and radiative forcing ∆R of CO2 and land ice albedo for the last 800,000 years we show that this state-dependency of S is only found if ∆Tg is based on reconstructions, and not when ∆Tg is based on model simulations. Furthermore, during times of decreasing obliquity (periods of land-ice sheet growth and sea level fall) the multi-millennial component of reconstructed ∆Tg is diverging from atmospheric CO2, while in simulations both variables vary more synchronously. For a reconstruction-based extrapolation of S to the future we eliminate these periods due to an expected sea level rise. Consequently, S determined from proxy-based reconstructions without these data with strong ∆Tg-CO2 divergence is less state-dependent or even constant (state-independent), and yields into an equilibrium warming for 2 × CO2 of 1.9–3.8 K.