  Extremely low-mass white dwarf (ELM WD; $M_{\star}$ $\lesssim$ $0.18-0.20\ $
$M_{\odot}$) stars are thought to be formed in binary systems via stable or
unstable mass transfer. Although stable mass transfer predicts the formation of
ELM WDs with thick hydrogen (H) envelopes, and hence characterized by dominant
residual nuclear burning along the cooling branch, the formation of ELM WDs
with thinner H envelopes from unstable mass loss cannot be discarded. We
compute new evolutionary sequences for helium (He) core WD stars with thin H
envelope with the main aim of assessing the lowest $T_{\rm eff}$ that could be
reached by this type of stars. We generate a new grid of evolutionary sequences
of He core WD stars with thin H envelope in the mass range from $0.1554$ to
$0.2025 M_{\odot}$, and assess the changes in both the cooling times and
surface gravity induced by a reduction of the H envelope. We also determine,
taking into account the predictions of progenitor evolution, the lowest $T_{\rm
eff}$ reached by the resulting ELM WDs. We find that a slight reduction in the
H envelope yields a significant increase in the cooling rate of ELM WDs.
Because of this, ELM WDs with thin H envelope could cool down to $\sim 2500\
$K, in contrast with their canonical counterparts that cool down to $\sim 7000\
$K. In addition, we find that a reduction of the thickness of the H envelope
increases markedly the surface gravity ($g$) of these stars. If ELM WDs are
formed with thin H envelopes, they could be detected at very low $T_{\rm eff}$.
The detection of such cool ELM WDs would be indicative that they were formed
with thin H envelopes, thus opening the possibility of placing constraints to
the possible mechanisms of formation of this type of stars. Last but not least,
the increase in $g$ due to the reduction of the H envelope leads to
consequences in the spectroscopic determinations of these stars.
