Artificial spin ices provide a controlled platform for investigating diverse physical phenomena, such as geometric frustration, magnetic monopoles, and phase transitions, via deliberate design. Here, we introduce a novel approach by developing artificial spin ice on the surfaces of a three-dimensional cube, which leads to emergent geometric frustration mediated by topologically protected domain walls, distinct from its flat counterparts. These domain walls connect vertices at the corners of cube that acting as intrinsic topological defects. Utilizing Monte Carlo simulations, we observe robust, topologically protected correlations among the intrinsic topological defects, regardless of their spatial separation. Our findings demonstrate that three-dimensional surfaces can unveil emergent properties absent in flat architectures.