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The equilibrium climate sensitivity (ECS) in the latest version of CNRM climate model, CNRM‐CM6‐1, and in its high‐resolution counterpart, CNRM‐CM6‐1‐HR, is significantly larger than in the previous version (CNRM‐CM5.1). The traceability of this climate sensitivity change is investigated using coupled ocean‐atmosphere model climate change simulations. These simulations show that the increase in ECS is the result of changes in the atmospheric component. A particular attention is paid to the method used to decompose the equilibrium temperature response difference, by using a linearized decomposition of the individual radiative agents diagnosed by a radiative kernel technique. The climate sensitivity increase is primarily due to the cloud radiative responses, with a predominant contribution of the tropical longwave response (including both feedback and forcing adjustment) and a significant contribution of the extratropical and tropical shortwave feedback changes. A series of stand‐alone atmosphere experiments is carried out to quantify the contributions of each atmospheric development to this difference between CNRM‐CM5.1 and CNRM‐CM6‐1. The change of the convection scheme appears to play an important role in driving the cloud changes, with a large effect on the tropical longwave cloud feedback change. Plain Language Summary: The global equilibrium temperature change in response to a doubling of the atmospheric carbon dioxide concentration is an important characteristic of the climate system known as the equilibrium climate sensitivity (ECS). Many climate models contributing to CMIP6 (Coupled Model Intercomparison Project phase 6) have a larger ECS than their CMIP5 predecessors. Here, we investigate the origins of this increase for the CNRM model and its high‐resolution version. We find that it primarily results from changes in the atmospheric component, in particular in the convection scheme, through its impact on the cloud radiative responses. Key Points: The CNRM climate models contributing to Coupled Model Intercomparison Project phase 6 have a larger climate sensitivity than their CMIP5 predecessorThe climate sensitivity increase is the result of changes in the atmospheric component, through the dominant role of tropical cloud changesThe new convection scheme appears to play an important role in driving the cloud changes [ABSTRACT FROM AUTHOR]