Proto-neutron stars forming a few seconds after core-collapse supernovae are hot and dense environments where hyperons can be efficiently produced by weak processes. By making use of various state-of-the-art supernova simulations combined with the proper extensions of the equations of state including $\Lambda$ hyperons, we calculate the cooling of the star induced by the emission of dark particles $X^0$ through the decay $\Lambda\to n X^0$. Comparing this novel energy-loss process to the neutrino cooling of SN 1987A allows us to set a stringent upper limit on the branching fraction, BR$(\Lambda\to n X^0)\leq 8\times10^{-9}$, that we apply to massless dark photons and axions with flavor-violating couplings to quarks. We find that the new supernova bound can be orders of magnitude stronger than other limits in dark-sector models.