  We study the thermal structure of the neutral atomic (H {\small I})
interstellar medium across a wide range of metallicities, from supersolar down
to vanishing metallicity, and for varying UV intensities and cosmic-ray
ionization rates. We calculate self-consistently the gas temperature and
species abundances (with a special focus on the residual H$_2$), assuming
thermal and chemical steady-state. For solar metallicity, $Z' \equiv 1$, we
recover the known result that there exists a pressure range over which the gas
is multiphased, with the warm ($\sim 10^4$ K, WNM) and cold ($\sim 100$ K, CNM)
phases coexisting at the same pressure. At a metallicity $Z' \approx 0.1$, the
CNM is colder (compared to $Z'=1$) due to the reduced efficiency of
photoelectric heating. For $Z' \lesssim 0.1$, cosmic-ray ionization becomes the
dominant heating mechanism and the WNM-to-CNM transition shifts to ever
increasing pressure/density as the metallicity is reduced. For metallicities
$Z' \lesssim 0.01$, H$_2$ cooling becomes important, lowering the temperature
of the WNM (down to $\approx 600$ K), and smoothing out the multiphase
phenomenon. At vanishing metallicities, H$_2$ heating becomes effective and the
multiphase phenomenon disappears entirely. We derive analytic expressions for
the critical densities for the warm-to-cold phase transition in the different
regimes, and the critical metallicities for H$_2$ cooling and heating. We
discuss potential implications on the star-formation rates of galaxies and
self-regulation theories.
