  If the early universe is dominated by an energy density which evolves other
than radiation-like the normal Hubble-temperature relation $H\propto T^2$ is
broken and dark matter relic density calculations in this era can be
significantly different. We first highlight that for a population of states
$\phi$ sourcing an initial expansion rate of the form $H\propto T^{2+n/2}$ for
$n\geq-4$, during the period of appreciable $\phi$ decays the evolution
transitions to $H\propto T^4$. The decays of $\phi$ imply a source of entropy
production in the thermal bath which alters the Boltzmann equations and impacts
the dark matter relic abundance. We show that the form of the initial expansion
rate leaves a lasting imprint on relic densities established while $H\propto
T^4$ since the value of the exponent $n$ changes the temperature evolution of
the thermal bath. In particular, a dark matter relic density set via freeze-in
or non-thermal production is highly sensitive to the temperature dependance of
the initial expansion rate. This work generalises earlier studies which assumed
initial expansion rates due to matter or kination domination.
