The ocean dominates the planetary heat budget and takes thousands of years to equilibrate to perturbed surface conditions, yet those long time scales are poorly understood. Here we analyze the ocean response over a range of forcing levels and time scales in a climate model of intermediate complexity and in the CMIP5 model suite. We show that on century to millennia time scales the response time scales, regions of anomalous ocean heat storage, and global thermal expansion depend nonlinearly on the forcing level and surface warming. As a consequence, it is problematic to deduce long-term from short-term heat uptake or scale the heat uptake patterns between scenarios. These results also question simple methods to estimate long-term sea level rise from surface temperatures, and the use of deep sea proxies to represent surface temperature changes in past climate.