Single atom catalysts (SACs) have aroused keen interests recently because of their superior catalytic performance and high atomic utilization. In this work, CO oxidation over the single Cu atom supported on two kinds of p-type doped (hole-doped and C-doped) hexagonal boron nitride (called Cu/Vb-BN and Cu/CBN hereafter) monolayer was systemically studied by DFT calculations. The good stability of the two SACs derives from obvious electrons transfer and orbital hybridization between Cu atom and the support. Both of the SACs follow the termolecular Eley-Rideal (TER) mechanism with OCOOCO* as intermediate. The barrier of rate-limited step is only 0.45 eV for Cu/Vb-BN, meaning its potential applications for low-temperature CO oxidation, whereas 0.90 eV for Cu/CBN, indicating it more suitable for higher temperature reaction because of enhanced adsorption strength of reactant and structural stability. The results of this work provide a theoretical evidence that the low-cost Cu supported on p-type h-BN monolayer catalyst could be a promising candidate for oxidation reactions.