  From five-dimensional $N=4$ matter-coupled gauged supergravity, smooth
time-dependent cosmological solutions, connecting a $dS_{5-d}\times H^d$ (with
$d=2,3$) spacetime at early times to a $dS_5$ spacetime at late times, are
presented. The solutions are derived from the second-order equations of motion
arising from all the gauged theories that can admit $dS_5$ solutions. There are
eight such theories constructed from gauge groups of the form $SO(1,1)\times
G_{nc}$ and $SO(1,1)^{(n)}_\text{diag}\times G_{nc}$, with $n=2,3$, where
$G_{nc}$ is a non-compact gauge factor whose compact part must be embedded
entirely in the matter symmetry group of 5D matter-coupled supergravity.
Furthermore, we analyze how the cosmological solutions and their corresponding
$dS_5$ vacua cannot arise from the first-order equations that solve the
second-order field equations.
