We compare the low-redshift (z ≃ 0.1) Lyα forest from hydrodynamical simulations with data from the Cosmic Origins Spectrograph. We find the tension between the observed number of lines with $b$-parameters in the range of 25–45 km s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ and the predictions from simulations that incorporate either vigorous feedback from active galactic nuclei or that exclude feedback altogether. The gas in these simulations is, respectively, either too hot to contribute to the Lyα absorption or too cold to produce the required linewidths. Matching the observed $b$-parameter distribution therefore requires feedback processes that thermally or turbulently broaden the absorption features without collisionally (over)ionizing hydrogen. This suggests that the Lyα forest $b$-parameter distribution is a valuable diagnostic of galactic feedback in the low-redshift Universe. We furthermore confirm that the low-redshift Lyα forest column density distribution is better reproduced by an ultraviolet background with the H$I$ photoionization rate a factor of 1.5–3 higher than predicted by Haardt and Madau.