  Within the framework of canonical type-I seesaw, a feebly interacting massive
particle (FIMP) $\chi$ is introduced as a dark matter candidate. The
leptogenesis mechanism and dark matter relic density share a common origin via
decays of Majorana neutrinos $N$. Provided an additional species $\varphi$
whose energy density red-shifts as $\rho_{\varphi}\propto a^{-(4+n)}$, the
Hubble expansion rate is larger than the standard scenario, i.e., the Universe
expands faster. The consequences of such a fast expanding Universe on
leptogenesis as well as FIMP dark matter are investigated in detail. We
demonstrate a significant impact on the final baryon asymmetry and dark matter
abundance due to the existence of $\varphi$ for the strong washout scenario.
While for the weak washout scenario, the effects of FEU are relatively small.
We introduce scale factors $F_L$ and $F_\chi$ to describe the corresponding
effects of FEU. A semi-analytical approach to derive the efficiency factors
$\eta_L$ and $\eta_\chi$ in FEU is also discussed. The viable parameter space
for success thermal leptogenesis and correct FIMP DM relic density is obtained
for standard cosmology and FEU. Our results show that it is possible to
distinguish different cosmology scenarios for strong washout cases.
