  We examine the column density distribution function (CDDF) and Doppler
parameter distribution from hydrodynamical simulations and Cosmic Origins
Spectrograph (COS) observations of the Lyman-alpha forest at redshift $0\leq
z\leq 0.2$. Allowing for a factor of two uncertainty in the metagalactic HI
photoionisation rate, our hydrodynamical simulations are in good agreement
($1$-$1.5\sigma$) with the shape and amplitude of the observed CDDF at HI
column densities $10^{13.3}\rm\,cm^{-2}\leq N_{\rm HI}\leq
10^{14.5}\rm\,cm^{-2}$. However, the Doppler widths of the simulated lines
remain too narrow with respect to the COS data. We argue that invoking AGN
feedback does not resolve this discrepancy. We also disfavour enhanced
photoheating rates as a potential solution, as this requires an unphysically
hard UV background spectrum. If instead appealing to a non-canonical source of
heating, an additional specific heat injection of $u \lesssim
6.9\rm\,eV\,m_{\rm p}^{-1}$ is required at $z\lesssim 2.5$ for gas that has
$N_{\rm HI}\simeq 10^{13.5}\rm\,cm^{-2}$ by $z=0.1$. Alternatively, there may
be an unresolved line of sight turbulent velocity component of $v_{\rm
turb}\lesssim 8.5\rm\,km\,s^{-1}(N_{\rm HI}/10^{13.5}\rm\,cm^{-2})^{0.21}$ for
the coldest gas in the diffuse IGM.
