<p>This work develops a novel energy-stable finite difference method that is of arbitrary high-order accuracy in principle for high-dimensional Allen-Cahn equations with mixed boundary conditions. The matched interface and boundary (MIB) method is employed to efficiently generate fictitious values outside of the domain, enabling the treatment of Dirichlet, Neumann, Robin boundary conditions and their combinations. This approach ensures an accurate approximation of derivative jumps at the domain boundaries, allowing for the correction of high-order central differences across all Cartesian grid nodes. Introducing the derivative jump terms as auxiliary variables leads to an augmented system that enables efficient FFT-based inversion of the discrete Laplacian via the Schur complement. A stabilized semi-implicit Crank-Nicolson discretization is employed to ensure second-order temporal accuracy, unconditional stability and energy dissipation. Numerical examples validate the accuracy and stability and demonstrate the efficiency of the proposed method.</p>