Climate policies are often designed to limit global mean surface warming to a certain threshold (e.g., 1.5°C), and coupled model intercomparison projects are similarly designed to span a range of future final greenhouse gas concentrations. However, what the global climate looks like at such targets may depend not only on the level of global warming, but also on how quickly greenhouse gases are increased to achieve that level. This raises the question: for a fixed amount of greenhouse gas increase, how does the global climate system respond to different rates of increase? In this talk, I will address this question for the Atlantic Meridional Overturning Circulation (AMOC) and Arctic surface temperatures, two components of the Earth system that may be particularly sensitive to anthropogenic climate change. Using a series of multi-millennial simulations in the low-resolution configuration of CESM1 in which atmospheric CO2 is doubled over different amounts of time, I show that the AMOC weakens substantially more under faster rates of CO2 increase due to a positive feedback with Arctic sea ice meltwater. This “rate-sensitive” AMOC behavior, in combination with amplifying radiative feedbacks, leads to a multi-centennial suppression of the signal of Arctic amplification when CO2 is increased sufficiently fast. Together, these results suggest that the rate of greenhouse gas increase---separately from the magnitude of increase--- may itself be an important control on the climate that merits further study in coupled climate models.