  Fermion dynamics distinguishes spacetimes having the same metric
$g_{\mu\nu}$, but different tetrads $e_{\mu a}$, and in particular, it
distinguishes a lapse with negative sign, $N<0$.[1] Here we show that the
quasiequilibrium thermodynamic state may exist, in which the region with $N<0$
has negative local temperature $T({\bf r})<0$, while the global Tolman
temperature $T_0$ remains positive. For bosons, only $N^2$ matters. However, if
bosons are composite, they may inherit the negative $T({\bf r})$ from the
fermions, and thus they may distinguish the spacetimes with positive and
negative lapse functions via thermodynamics.
