  We report evidence for an anti-correlation between spin temperature $T_s$ and
metallicity [Z/H], detected at $3.6 \sigma$ significance in a sample of 26
damped Lyman-$\alpha$ absorbers (DLAs) at redshifts $0.09 < z < 3.45$. The
anti-correlation is detected at $3 \sigma$ significance in a sub-sample of 20
DLAs with measured covering factors, implying that it does not stem from low
covering factors. We obtain $T_s = (-0.68 \pm 0.17) \times {\rm [Z/H]} + (2.13
\pm 0.21)$ from a linear regression analysis. Our results indicate that the
high $T_s$ values found in DLAs do not arise from differences between the
optical and radio sightlines, but are likely to reflect the underlying gas
temperature distribution. The trend between $T_s$ and [Z/H] can be explained by
the larger number of radiation pathways for gas cooling in galaxies with high
metal abundances, resulting in a high cold gas fraction, and hence, a low spin
temperature. Conversely, low-metallicity galaxies have fewer cooling routes,
yielding a larger warm gas fraction and a high $T_s$. Most DLAs at $z>1.7$ have
low metallicities, [Z/H] $< -1$, implying that the HI in high-$z$ DLAs is
predominantly warm. The anti-correlation between $T_s$ and [Z/H] is consistent
with the presence of a mass-metallicity relation in DLAs, suggested by the
tight correlation between DLA metallicity and the kinematic widths of metal
lines. Most high-$z$ DLAs are likely to arise in galaxies with low masses
($M_{\rm vir} < 10^{10.5} M_\odot$), low metallicities ([Z/H]$< -1$, and low
cold gas fractions.
