We report on the comparative study of AlGaN-GaN Heterostructure Field Effect Transistors (HFETs) fabricated on sapphire and conducting 6H-SiC substrates. The SiC substrates dramatically decrease the thermal impedance compared to similar devices grown on sapphire. These conducting substrates also act as second gates, whose performance depends on the substrate conductivity type. The measured transconductance values for these bottom gates are on the order of 50 mS/mm. The devices on SiC are capable to operate at DC powers up to 0.8 MW/cm(2). Their measured thermal impedance is approximately 2 degrees C mm/W (about 20 times smaller than for typical GaAs power FETs.) We conclude that the AlGaN/GaN HFETs on SiC substrates combine superior thermal properties of the two-dimensional electrons in GaN with excellent thermal properties of silicon carbide. Therefore, these devices have a great potential for power microwave, millimeter wave, and, switching applications.