We investigate carbon single-atom and pair defects at the SiC/SiO2 interface as candidate defects for the density of defect states in the SiC band gap. In order to accurately describe the electronic defect levels with respect to the SiC band edges, we use a hybrid density functional which reproduces the experimental band gap of SiC. The carbon pair defect consisting of two neighboring sp(2) hybridized carbon atoms is modeled in various configurations within a SiC/SiO2 model interface showing good structural parameters and an oxide density typical of amorphous SiO2. The carbon pair defect is found to contribute to the density of defect states not only in the lower and/or mid band gap of SiC, but also in the upper band gap, in contrast with previous studies. The carbonpair defect is also investigated via molecular models to achieve insight into the energy range spanned by its defect levels when the relative orientation of its sp(2) hybridization planes and the chemical nature of its neighbors are varied. Carbon single-atom defects on the oxide side of the interface are also modeled and found to contribute to the defect density in the band gap in a similar way as carbonpair defects. Comparison to the experimental defect density suggests that defects involving one or two carbon atoms cannot account for the high defect density observed in the vicinity of the SiC conduction band.