Efficient and compact power conversion is a key requirement to achieve sustainable electrification of our society. GaN-based power devices offer major benefits compared to their silicon counterparts in terms of efficiency and integration but suffer from severe thermal challenges. Self-heating negatively impacts device performance and reliability, and the lateral integration of power devices and logic on a single integrated circuit cannot be fully exploited without novel cooling methods. In this work, we show a method to integrate microchannel cooling inside the silicon substrate of an off-the-shelf GaN-on-Si power IC and achieve a 25x-reduction on thermal resistance compared to forced air cooling. We investigate measurement techniques to measure the device temperature when no direct physical or optical connection can be made to the chip. A side-by-side comparison in electrical and thermal performance between conventional (forced) air-cooling shows that the integration of liquid cooling reduces the negative effects of self-heating on electrical performance while significantly improving its maximum current capability. The results show a first step toward making high-performance power converters with state-of-the-art thermal performance.