  We consider a modification to the standard cosmological history consisting of
introducing a new species $\phi$ whose energy density red-shifts with the scale
factor $a$ like $\rho_\phi \propto a^{-(4+n)}$. For $n>0$, such a red-shift is
faster than radiation, hence the new species dominates the energy budget of the
universe at early times while it is completely negligible at late times. If
equality with the radiation energy density is achieved at low enough
temperatures, dark matter can be produced as a thermal relic during the new
cosmological phase. Dark matter freeze-out then occurs at higher temperatures
compared to the standard case, implying that reproducing the observed abundance
requires significantly larger annihilation rates. Here, we point out a
completely new phenomenon, which we refer to as $\textit{relentless}$ dark
matter: for large enough $n$, unlike the standard case where annihilation ends
shortly after the departure from thermal equilibrium, dark matter particles
keep annihilating long after leaving chemical equilibrium, with a significant
depletion of the final relic abundance. Relentless annihilation occurs for $n
\geq 2$ and $n \geq 4$ for s-wave and p-wave annihilation, respectively, and it
thus occurs in well motivated scenarios such as a quintessence with a kination
phase. We discuss a few microscopic realizations for the new cosmological
component and highlight the phenomenological consequences of our calculations
for dark matter searches.
