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Abstract Perovskite oxides offer tremendous potential for applications in information storage and energy conversion, owing to a subtle interplay between their spin, charge, orbital and lattice degrees of freedom. Here, we further expand the possible range of perovskite oxides operation towards the fields of thermal management and thermal computing by exploiting an exceptional synergy between different ferroic orders. We propose dynamical control of the heat flow in a distinctive family of perovskite oxides obtained via the application of small electric (~10 kV/cm) and/or magnetic (~1 T) fields. Based on first-principles simulations, we predict a relative heat conductivity variation of ~100% in SrMnO3 thin films near room temperature resulting from a phase transition that involves huge changes in both the magnetization and electric polarization. The disclosed giant multiphononic effects are fundamentally caused by anharmonic spin-phonon couplings that strongly influence the mean lifetime of phonons.