We demonstrate theoretically that the thermal Hall effect of magnons in collinear antiferromagnetic insulators is an indicator of magnetic and topological phase transitions in the magnon spectrum. The transversal heat current of magnons caused by a thermal gradient is calculated for an antiferromagnet on a honeycomb lattice. An applied magnetic field drives the system from the antiferromagnetic phase via a spin-flop phase into the field-polarized phase. Besides these magnetic phase transitions we find topological phase transitions within the spin-flop phase. Both types of transitions manifest themselves in prominent and distinguishing features in the thermal conductivities; depending on the temperature, the conductivity changes by several orders of magnitude, providing a tool to discern experimentally the two types of phase transitions. We include numerical results for the van der Waals magnet MnPS$_3$.
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