This study concentrates on the Allen–Cahn equation with logarithmic free energy, thereby proposing a second–order accurate finite difference scheme. The core design of the scheme adopts a linearized modified version of the classical leapfrog scheme for temporal discretization, combined with central differencing for spatial discretization. To further improve the numerical stability, a second–order stabilization term is systematically integrated into the discrete framework. A theoretical analysis reveals that under appropriate constraints on the time step size and stabilization parameter, the numerical solution strictly adheres to the maximum bound principle. A comprehensive stability analysis is performed in the maximum norm, and the corresponding error estimates are rigorously derived. Finally, numerical experiments are conducted to verify the correctness and effectiveness of the theoretical results.