We study the effects of modifying the expansions history of the Universe on dark matter freeze-out. We derived a modified Boltzmann equation for freeze-out for an arbitrary energy density in the early Universe and provide an analytic approach using some approximations. We then look at the required thermally averaged cross sections needed to obtain the correct relic density for the specific case where the energy density consists of radiation plus one extra component which cools faster. We compare our analytic approximation to numerical solutions. We find that it gives reasonable results for most of the parameter space explored, being at most a factor of order 1 away from the measured value. We find that if the new contribution to the energy density is comparable to the radiation density, then a much smaller cross section for dark matter annihilation is required. This would lead to weak scale dark matter being much more difficult to detect and opens up the possibility that much heavier dark matter could undergo freeze-out without violating perturbative unitarity.