Global energy budget constraints suggest an equilibrium climate sensitivity around 2 °C, which is lower than estimates from palaeoclimate reconstructions, process-based observational analyses, and global climate model simulations. A key assumption is that the climate sensitivity inferred today also applies to the distant future. Yet, global climate models robustly show that feedbacks vary over time, with a strong tendency for climate sensitivity to increase as equilibrium is approached. Here I consider the implications of inconstant climate feedbacks for energy budget constraints on climate sensitivity. I find that the long-term value of climate sensitivity is, on average, 26% above that inferred during transient warming within global climate models, with a larger discrepancy when climate sensitivity is high. Moreover, model values of climate sensitivity inferred during transient warming are found to be consistent with energy budget observations, indicating that the models are not overly sensitive. Using model-based estimates of how climate feedbacks will change in the future, in conjunction with recent energy budget constraints, produces a current best estimate of equilibrium climate sensitivity of 2.9 °C (1.7-7.1 °C, 90% confidence). These findings suggest that climate sensitivity estimated from global energy budget constraints is in agreement with values derived from other methods and simulated by global climate models.