Integrated optomechanical systems are a leading platform for manipulating, sensing, and distributing quantum information, but are limited by residual optical heating. Here, we demonstrate a two-dimensional optomechanical crystal (OMC) geometry with increased thermal anchoring and a mechanical mode at 7.4 GHz, well aligned with the operation range of cryogenic microwave hardware and piezoelectric transducers. The eight times better thermalization than current one-dimensional OMCs, large optomechanical coupling rates, g