We present a thorough investigation of surface deformation and thermal properties of high-damage threshold large-area semiconductor saturable absorber mirrors (SESAMs) designed for kilowatt average power laser oscillators. We compare temperature rise, thermal lensing, and surface deformation of standard SESAM samples and substrate-removed SESAMs contacted using different techniques. We demonstrate that for all cases the thermal effects scale linearly with the absorbed power, but the contacting technique critically affects the strength of the temperature rise and the thermal lens of the SESAMs (i.e. the slope of the linear change). Our best SESAMs are fabricated using a novel substrate-transfer direct bonding technique and show excellent surface flatness (with non-measureable radii of curvature (ROC), compared to astigmatic ROCs of up to 10 m for standard SESAMs), order-of-magnitude improved heat removal, and negligible deformation with absorbed power. This is achieved without altering the saturation behavior or the recovery parameters of the samples. These SESAMs will be a key enabling component for the next generation of kilowatt-level ultrafast oscillators.