Deep traps in semi-insulating (SI) or high-resistivity C-doped GaN grown by metal-organic chemical-phase deposition or molecular-beam epitaxy have been studied by thermally stimulated current (TSC) spectroscopy. Incorporation of carbon in GaN introduces CN acceptors, resulting in compensation and formation of SI-GaN; however, as [C] increases in the GaN samples, both resistivity and activation energy of the dark current decrease. In the GaN samples with low [C], we find at least six TSC traps: B (0.61 eV), Bx (0.50 eV), C1 (0.44 eV), C (0.32 eV), D (0.23 eV), and E (0.16 eV), all of which are very similar to electron traps typically found in n-type GaN by deep level transient spectroscopy (DLTS). However, in the GaN sample with the highest [C], both traps E and B are suppressed, and instead, trap Bx appears. Based on DLTS studies of electron-irradiated and plasma-etched GaN samples, we believe that traps E, D and C are related to VN, and that trap B is probably related to VGa, in the form of complexes such as VGa-ON. As [C] increases, CGa donors become more favorable, and the transition of trap B to trap Bx may suggest that CGa related complexes are forming. In comparison with lightly C-doped GaN, heavily C-doped GaN sample exhibits very strong PPC at 83 K. We show that the PPC in both cases can be simply explained by the thermal emission of carriers from shallower traps.