We are in a period of relatively rapid climate change. This poses challenges for individual species and threatens the ecosystem services that humanity relies upon. Temperature is a key stressor. In a warming climate, individual organisms may be able to shift their thermal optima through phenotypic plasticity. However, such plasticity is unlikely to be sufficient over the coming centuries. Resilience to warming will also depend on how fast the distribution of traits that define a species can adapt through other methods, in particular through redistribution of the abundance of variants within the population and through genetic evolution. In this paper, we use a simple theoretical ‘trait diffusion’ model to explore how the resilience of a given species to climate change depends on the initial trait diversity (biodiversity), the trait diffusion rate (mutation rate), and the lifetime of the organism. We estimate theoretical dangerous rates of continuous global warming that would exceed the ability of a species to adapt through trait diffusion, and therefore lead to a collapse in the overall productivity of the species. As the rate of adaptation through intraspecies competition and genetic evolution decreases with species lifetime, we find critical rates of change that also depend fundamentally on lifetime. Dangerous rates of warming vary from 1°C per lifetime (at low trait diffusion rate) to 8°C per lifetime (at high trait diffusion rate). We conclude that rapid climate change is liable to favour short-lived organisms (e.g. microbes) rather than longer-lived organisms (e.g. trees).