Mesoscale ocean temperature anomalies modify a tropical cyclone (TC). Through a modeling study we show that, while the maximum wind speed is rapidly restored after the TC passes a warm- or cold- (eddy size) sea surface temperature (SST) anomaly, the storm size changes are more significant and persistent. The radius of gale force winds and integrated kinetic energy (IKE) can change by more than 10% per degree and this endures several days after crossing an SST anomaly. These properties have a long memory of the impact from the ocean fluxes and depend on the integrated history of SST exposure. They are found to be directly proportional to the storm total precipitation. Accurate continuous forecast of the SST along the track may therefore be of central importance to improving predictions of size and IKE, while instantaneous local SST near the TC core is more important for the forecast of maximum wind speed.