Many dark matter candidates, including asymmetric Weakly Interacting Massive Particles (WIMPs) and sterile neutrinos, are produced in the very early Universe, prior to Big Bang Nucleosynthesis (BBN). We show that the relic abundance of asymmetric WIMPs and sterile neutrinos can be very sensitive to the expansion rate of the Universe prior to BBN. In particular, we find that if the production of asymmetric WIMPs occurs during a non-standard cosmological phase, a larger WIMP annihilation cross section is required to produce the present dark matter density than if the WIMPs were produced during a standard, radiation dominated phase. Because of this, the present dark matter annihilation rate could be larger than that of symmetric dark matter produced in the standard cosmology. We also show that if the production of sterile neutrinos occurs during a non-standard cosmological phase, the relic number density of sterile neutrinos could be reduced with respect to the number expected in the standard cosmology, consequently relaxing current bounds on active-sterile neutrino mixing. Finally, we examine whether low reheating temperature cosmologies are allowed by current Cosmic Microwave Background measurements. We find the allowed range of reheating temperatures using monomial and binomial inflationary potentials, and a variety of reheating models. We show that an inflationary model with a φ^1 potential and canonical reheating allows the possibility that dark matter could be produced during the reheating epoch, instead of when the Universe is radiation dominated.