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Abstract: Strategies for tuning the thermal conductivity of crystals by means of external fields are rare. Here, we predict the existence of large magnetophononic effects in materials that undergo antiferromagnetic (AFM) $\ensuremath{\leftrightarrow}$ ferromagnetic (FM) phase transitions, which allow for the modulation of the lattice heat conductivity, ${\ensuremath{\kappa}}_{L}$, via the application of magnetic fields. Specifically, by using first-principles methods we predict a large and anomalous ${\ensuremath{\kappa}}_{L}$ increase of $\ensuremath{\approx}40%$ for the metamagnetic phase transition occurring in bulk FeRh near room temperature. The disclosed magnetophononic effects are caused by large anharmonic spin-phonon couplings, namely, significant differences in the phase space of allowed phonon-phonon collision processes taking place in the respective AFM and FM phases.