Rapid, accurate prediction of spatio-temporal temperature fields is critical for process-aware design and control in metal additive manufacturing (AM). Conventional finite-element solvers, however, are too slow for iterative design or real-time monitoring. In this work, we present the Harmonic-Mapping Operator (HMO), a geometry-agnostic surrogate that learns the thermo-physical response of arbitrarily shaped parts without retraining for each new geometry. HMO separates geometric complexity from thermal dynamics by constructing a harmonic map that warps any build domain to a canonical domain (e.g., the unit square) while simultaneously transforming the heat source and scan trajectories. A neural operator trained once on this canonical domain solves the transient heat equation, and an inverse map then projects the predicted temperature field back to the original geometry. Because training occurs in a fixed coordinate system and geometric information is encoded as a metric tensor on the canonical domain, HMO generalizes naturally to unseen geometries. Benchmarked on diverse two-dimensional laser powder-bed fusion cases, HMO predicts full-layer thermal histories in milliseconds, achieving lower error and stronger geometric generalization than existing data-driven or physics-informed surrogates. These results show that harmonic mapping coupled with neural operators yields a single reusable surrogate with physics-level accuracy at real-time speeds, paving the way for in-situ optimization and closed-loop control in metal AM.