In this study, we investigated the operational characteristics of AlGaN/GaN high electron mobility transistors (HEMTs) by applying the copper-filled trench and via structures for improved heat dissipation. Therefore, we used a basic T-gate HEMT device to construct the thermal structures. To identify the heat flow across the device structure, a thermal conductivity model and the heat transfer properties corresponding to the GaN, SiC, and Cu materials were applied. Initially, we simulated the direct current (DC) characteristics of a basic GaN on SiC HEMT to confirm the self-heating effect on AlGaN/GaN HEMT. Then, to verify the heat sink effect of the copper-filled thermal structures, we compared the DC characteristics such as the threshold voltage, transconductance, saturation current, and breakdown voltage. Finally, we estimated and compared the lattice temperature of a two-dimensional electron gas channel, the vertical lattice temperature near the drain-side gate head edge, and the transient thermal analysis for the copper-filled thermal trench and via structures. Through this study, we could optimize the operational characteristics of the device by applying an effective heat dissipation structure to the AlGaN/GaN HEMT.