We theoretically study the thermal relaxation of many-body systems under the action of oscillating external fields. When the magnitude or the orientation of a field is modulated around values where the pairwise heat-exchange conductances depend non-l
inearly on this field, we demonstrate that the time symmetry is broken during the evolution of temperatures over a modulation cycle. We predict that this asymmetry enables a pumping of heat which can be used to cool down faster the system. This effect is illustrated through different magneto-optical systems under the action of an oscillating magnetic field.
