  Recent work has suggested that an additional $\lesssim 6.9 \rm{\, eV}$ per
baryon of heating in the intergalactic medium is needed to reconcile
hydrodynamical simulations with Lyman-$\alpha$ forest absorption line widths at
redshift $z\simeq 0.1$. Resonant conversion of dark photon dark matter into low
frequency photons is a viable source of such heating. We perform the first
hydrodynamical simulations including dark photon heating and show that dark
photons with mass $m_{A'}\sim 8\times 10^{-14}\rm\,eV\,c^{-2}$ and kinetic
mixing $\epsilon \sim 5\times 10^{-15}$ can alleviate the heating excess. A
prediction of this model is a non-standard thermal history for underdense gas
at $z \gtrsim 3$.
