At low temperatures, a spin ice enters a Coulomb phase - a state with algebraic correlations and topologically constrained spin configurations. We show how analytical and numerical approaches for model spin-ice systems reveal a crossover between two Curie laws. One of these laws characterizes the high-temperature paramagnetic regime, while the other, which we call the "spin-liquid Curie law," characterizes the low-temperature Coulomb-phase regime, which provides implicit evidence that the topological sector fluctuates. We compare our theory with experiment for HO₂Ti₂O₇, where this process leads to a nonstandard temperature evolution of the bulk susceptibility and the wave-vector-dependent magnetic susceptibility, as measured by neutron scattering. Theory and experiment agree for bulk quantities and at large scattering wave vectors, but differences at small wave vectors indicate that the classical spin-ice states are not equally populated at low temperatures. More generally, the crossover appears to be a generic property of the emergent gauge field for a classical spin liquid, and it sheds light on the experimental difficulty of measuring a precise Curie-Weiss temperature in frustrated materials. The susceptibility at finite wave vectors is shown to be a local probe of fluctuations among topological sectors on varying length scales.