  Our ignorance of the period between the end of inflation and the beginning of
Big Bang Nucleosynthesis limits our understanding of the origins and evolution
of dark matter. One possibility is that the Universe's energy density was
dominated by a fast-rolling scalar field while the radiation bath was hot
enough to thermally produce dark matter. We investigate the evolution of the
dark matter density and derive analytic expressions for the dark matter relic
abundance generated during such a period of kination. Kination scenarios in
which dark matter does not reach thermal equilibrium require $\langle \sigma v
\rangle < 2.7\times 10^{-38} \,\mathrm{cm^3\,s^{-1}}$ to generate the observed
dark matter density while allowing the Universe to become radiation dominated
by a temperature of $3 \, \mathrm{MeV}$. Kination scenarios in which dark
matter does reach thermal equilibrium require $\langle \sigma v \rangle >
3\times 10^{-26} \,\mathrm{cm^3\,s^{-1}}$ in order to generate the observed
dark matter abundance. We use observations of dwarf spheroidal galaxies by the
Fermi Gamma-Ray Telescope and observations of the Galactic Center by the High
Energy Stereoscopic System to constrain these kination scenarios. Combining the
unitarity constraint on $\langle \sigma v \rangle$ with these observational
constraints sets a lower limit on the temperature at which the Universe can
become radiation dominated following a period of kination if ${\langle \sigma v
\rangle > 3\times 10^{-31} \,\mathrm{cm^3\,s^{-1}}}$. This lower limit is
between ${0.05 \, \mathrm{GeV}}$ and ${1 \, \mathrm{GeV}}$, depending on the
dark matter annihilation channel.
