Climate-related risks are dependent not only on the warming trend from GHGs, but also on the variability about the trend. However, assessment of the impacts of climate change tends to focus on the ultimate level of global warming1, only occasionally on the rate of global warming, and rarely on variability about the trend. Here we show that models that are more sensitive to GHGs emissions (that is, higher equilibrium climate sensitivity (ECS)) also have higher temperature variability on timescales of several years to several decades2. Counter-intuitively, high-sensitivity climates, as well as having a higher chance of rapid decadal warming, are also more likely to have had historical `hiatus' periods than lower-sensitivity climates. Cooling or hiatus decades over the historical period, which have been relatively uncommon, are more than twice as likely in a high-ECS world (ECS = 4.5 K) compared with a low-ECS world (ECS = 1.5 K). As ECS also affects the background warming rate under future scenarios with unmitigated anthropogenic forcing, the probability of a hyper-warming decade—over ten times the mean rate of global warming for the twentieth century—is even more sensitive to ECS.