We have measured low-temperature variation of resistance and nonlinear current-voltage behavior in antidot graphene in the vicinity of the charge neutrality point. The data are found to be consistent with the manifestations of a variable-range hopping electronic density of states (DOS) with a small hard gap of ∼1 meV around the Fermi level, in conjunction with a parallel tunneling conduction channel that exists at the center of the gap. The hard gap is confirmed by the appearance of a low-conductive plateau at low-bias electric field, whereas the parallel tunneling conduction channel, with temperature-independent conductance, is manifest through the nonlinear electric field variation. Unified good agreement between the temperature and electric field dependencies of conductance, for both channels, is obtained with the predictions of a proposed DOS model. An increase in the gap size with applied magnetic field is observed.