Simulated cooling is a robust method for preparing low-energy states of many-body Hamiltonians on near-term quantum simulators. In such schemes, a subset of the simulator's spins (or qubits) are treated as a "bath" that extracts energy and entropy from the system of interest. However, such protocols are inefficient when applied to systems whose excitations are highly nonlocal in terms of the microscopic degrees of freedom, such as topological phases of matter
such excitations are difficult to extract by a local coupling to a bath. We explore a route to overcome this obstacle by encoding the microscopic degrees of freedom into those of the quantum simulator in a nonlocal manner. To illustrate the approach, we show how to efficiently cool the ferromagnetic phase of the quantum Ising model, whose excitations are domain walls, via a "gauged cooling" protocol in which the Ising spins are coupled to a Z_{2} gauge field that simultaneously acts as a reservoir for removing excitations. We show that our protocol can prepare the ground states of the ferromagnetic and paramagnetic phases equally efficiently. The gauged cooling protocol naturally extends to (interacting) fermionic systems, where it is equivalent to cooling by coupling to a fermionic bath via single-fermion hopping.