The modified cosmology like quintessence model with kination phase predicted the Hubble expansion rate of the universe before Big Bang Nucleosynthesis is different from the standard cosmological scenario. The modified expansion rate leaves its imprint on the relic density of asymmetric dark matter. In this work, we review the calculation of relic density of asymmetric WIMP dark matter in the standard cosmological scenario and quintessence model with kination phase. Then we use the Planck data to find constraints on the annihilation cross section and the mass of the asymmetric dark matter in those models.