Relationships between climate variability and climate sensitivity are to be expected where the damping of a climatic anomaly is due to a change in the energy balance of the planet, such that the Fluctuation-Dissipation theorem heuristically applies [Leith, 1975]. A recent attempt to relate Equilibrium Climate Sensitivity (ECS) to global temperature variability over the historical period suggested a surprisingly tight emergent constraint on ECS [Cox et al., 2018]. However, the sensitivity-variability relationship in that study was partially hidden by anthropogenic forcing over the historical period. Here we examine instead CMIP5 control runs. These runs have no external forcing and therefore provide a much cleaner test of proposed links between internal variability and sensitivity. It has been noted before that there is a positive correlation between decadal temperature variability and climate sensitivity across climate models [Colman & Power, 2018]. Questions remained however as to how robust this relationship is across different model ensembles, what mechanisms are responsible for it, and whether it can be used as an emergent constraint on climate sensitivity. We examine the relationship between decadal variability and ECS using models of varying complexity, including CMIP5 control runs and a range of conceptual energy balance models for which analytical solutions are presented. Based on these results, a general mechanism becomes apparent and the shape of the relationship is determined to be more quadratic than linear. The nonlinearity has implications for using this relationship as an emergent constraint, where an incorrect assumption of linearity might lead to biased estimates. A further surprising implication of the study is that a slowdown in global warming does not necessarily imply that climate sensitivity is lower than previously estimated. Models with a higher sensitivity, but which broadly reproduce the long-term record of global warming, are actually more likely to have slow-down periods than models with lower sensitivity.