Cold atomic gases have proven capable of emulating a number of fundamental condensed matter phenomena including Bose-Einstein condensation, the Mott transition, Fulde-Ferrell-Larkin-Ovchinnikov pairing, and the quantum Hall effect. Cooling to a low enough temperature to explore magnetism and exotic superconductivity in lattices of fermionic atoms remains a challenge. We propose a method to produce a low temperature gas by preparing it in a disordered potential and following a constant entropy trajectory to deliver the gas into a nondisordered state which exhibits these incompletely understood phases. We show, using quantum Monte Carlo simulations, that we can approach the Néel temperature of the threedimensional Hubbard model for experimentally achievable parameters. Recent experimental estimates suggest the randomness required lies in a regime where atom transport and equilibration are still robust. [5,6]. Ultracold atomic gases offer the opportunity to emulate these fundamental issues using optical speckle [7,8], impurities [9], or a quasiperiodic optical lattice [10,11] to introduce randomness. In the bosonic case, the competition between strong interactions and strong disorder has been studied in the context of the elusive Bose glass phase [7,9,11], while for fermions, a recent experiment has explored disorder-induced localization in the three-dimensional (3D) Hubbard model of strongly interacting fermions [12].In this paper, we explore the thermodynamics of interacting, disordered systems and suggest that, in addition to studies of the many-body phenomena noted above, preparing a gas in a random potential might be exploited to cool the atoms. Specifically, we show using an unbiased numerical method that one can lower the temperature and access the regime with long-range magnetic order by adiabatically decreasing the randomness in the chemical potential or hopping energies of the Hubbard Hamiltonian. The achievement of new quantum phases in cold atom experiments largely depends on the reduction of the entropy per particle. The success of our proposal requires that the gas would have to be cooled (e.g. evaporatively) after the disorder is in place. We will return in the