  Dark photons in the MeV to GeV mass range are important targets for
experimental searches. We consider the case where dark photons $A'$ decay
invisibly to hidden dark matter $X$ through $A' \to XX$. For generic masses,
proposed accelerator searches are projected to probe the thermal target region
of parameter space, where the $X$ particles annihilate through $XX \to A' \to
\text{SM}$ in the early universe and freeze out with the correct relic density.
However, if $m_{A'} \approx 2m_X$, dark matter annihilation is resonantly
enhanced, shifting the thermal target region to weaker couplings. For $\sim
10\%$ degeneracies, we find that the annihilation cross section is generically
enhanced by four (two) orders of magnitude for scalar (pseudo-Dirac) dark
matter. For such moderate degeneracies, the thermal target region drops to weak
couplings beyond the reach of all proposed accelerator experiments in the
scalar case and becomes extremely challenging in the pseudo-Dirac case.
Proposed direct detection experiments can probe moderate degeneracies in the
scalar case. For greater degeneracies, the effect of the resonance can be even
more significant, and both scalar and pseudo-Dirac cases are beyond the reach
of all proposed accelerator and direct detection experiments. For scalar dark
matter, we find an absolute minimum that sets the ultimate experimental
sensitivity required to probe the entire thermal target parameter space, but
for pseudo-Dirac fermions, we find no such thermal target floor.
