Electrochemical production of hydrogen peroxide (H2O2) is a sustainable and environmentally benign process. The electrochemical oxygen reduction process (ORR) via a two electron pathway (2e- ORR) offers a practical method for on-site H2O2 generation. As an earth-abundant catalyst, the cobalt-nitrogen coordinated systems integrated into the carbon matrix (Co-NC) has caused wide attention for its high activity in 2e- ORR. Even though most of the reported Co-NC catalysts have classical planar Co-N4 coordination, axial coordination engineering has recently emerged as an effective way to control the active sites in the axial direction by using different coordination ligands. The structure-function link between the Co-N configuration of non-planar coordination and 2e- ORR activity is, however, not fully understood. An axial-N coordinated Co-N5 motif embedded in hierarchically porous graphite-3R carbon (Co-N5C) was effectively synthesized using a template-sacrificing method. The Co-N5C has a high selectivity for 2e- ORR and a high H2O2 molar production rate of up to 6.78 mol peroxide/gcatalyst/h in acidic media, both of which are better than its Co-N4 counterpart. DFT analyses demonstrate that axial-N ligands regulated the d-band center of the Co atom in the Co-N5C catalyst, inducing a shift in ΔG*OOH near the Sabatier volcano plot's peak (ΔG*OOH = 4.22 eV). This optimized the binding of the *OOH intermediate and then enhanced the protonation of *OOH to produce H2O2 more efficiently.