Given the Paris Agreement it is imperative there is a greater understanding of the consequences of limiting global warming to the target 1.5°C and 2°C levels above pre-industrial conditions. It is challenging to quantify changes across a small increment of global warming, so a pattern-scaling approach may be considered. Here we investigate the validity of such an approach by comprehensively examining how well local temperatures and warming trends in a 1.5°C world predict local temperatures at global warming of 2°C. Ensembles of transient coupled climate simulations from multiple models under different scenarios were compared and individual model responses were analyzed. For many places, the multimodel forced response of seasonal-average temperatures is approximately linear with global warming between 1.5°C and 2°C. However, individual model results vary and large contributions from non-linear changes in unforced variability or the forced response cannot be ruled out. In some regions, such as East Asia, models simulate substantially greater warming than is expected from linear scaling. Examining East Asia during boreal summer, we find increased warming in the simulated 2°C world relative to scaling up from 1.5°C is related to reduced anthropogenic aerosol emissions. Our findings suggest that where forcings other than those due to greenhouse gas emissions change, the warming experienced in a 1.5°C world is a poor predictor for local climate at 2°C of global warming. In addition to the analysis of the linearity in the forced climate change signal, we find that natural variability remains a substantial contribution to uncertainty at these low-warming targets.