We investigate the physical state of H I absorbing gas at low redshift (z = 0.25) using a subset of cosmological, hydrodynamic simulations from the OverWhelmingly Large Simulations project, focusing in particular on broad (bH I≥40 km s-1) H I Lyα absorbers (BLAs), which are believed to originate in shock-heated gas in the warm-hot intergalactic medium (WHIM). Our fiducial model, which includes radiative cooling by heavy elements and feedback by supernovae and active galactic nuclei, predicts that by z = 0.25 nearly 60 per cent of the gas mass ends up at densities and temperatures characteristic of the WHIM and we find that half of this fraction is due to outflows. The standard H I observables (distribution of H I column densities NH I, distribution of Doppler parameters bH I, bH I-NH I correlation) and the BLA line number density predicted by our simulations are in remarkably good agreement with observations. BLAs arise in gas that is hotter, more highly ionized and more enriched than the gas giving rise to typical Lyα forest absorbers. The majority of the BLAs arise in warm-hot [log (T/ K) ˜ 5] gas at low (log Δ < 1.5) overdensities. On average, thermal broadening accounts for at least 60 per cent of the BLA linewidth, which in turn can be used as a rough indicator of the thermal state of the gas. Detectable BLAs account for only a small fraction of the true baryon content of the WHIM at low redshift. In order to detect the bulk of the mass in this gas phase, a sensitivity at least one order of magnitude better than achieved by current ultraviolet spectrographs is required. We argue that BLAs mostly trace gas that has been shock heated and enriched by outflows and that they therefore provide an important window on a poorly understood feedback process.