For more than a decade, the target of keeping global warming below 2 °C has been a key focus of the international climate debate¹. In response, the scientific community has published a number of scenario studies that estimate the costs of achieving such a target^{2, 3, 4, 5}. Producing these estimates remains a challenge, particularly because of relatively well known, but poorly quantified, uncertainties, and owing to limited integration of scientific knowledge across disciplines⁶. The integrated assessment community, on the one hand, has extensively assessed the influence of technological and socio-economic uncertainties on low-carbon scenarios and associated costs^{2, 3, 4, 7}. The climate modelling community, on the other hand, has spent years improving its understanding of the geophysical response of the Earth system to emissions of greenhouse gases^{8, 9, 10, 11, 12}. This geophysical response remains a key uncertainty in the cost of mitigation scenarios but has been integrated with assessments of other uncertainties in only a rudimentary manner, that is, for equilibrium conditions^{6, 13}. Here we bridge this gap between the two research communities by generating distributions of the costs associated with limiting transient global temperature increase to below specific values, taking into account uncertainties in four factors: geophysical, technological, social and political. We find that political choices that delay mitigation have the largest effect on the cost–risk distribution, followed by geophysical uncertainties, social factors influencing future energy demand and, lastly, technological uncertainties surrounding the availability of greenhouse gas mitigation options. Our information on temperature risk and mitigation costs provides crucial information for policy-making, because it clarifies the relative importance of mitigation costs, energy demand and the timing of global action in reducing the risk of exceeding a global temperature increase of 2 °C, or other limits such as 3 °C or 1.5 °C, across a wide range of scenarios.