Fermion dynamics distinguishes spacetimes having the same metric g μν , but different tetrads e μa , and in particular, it distinguishes a lapse with negative sign, N < 0 (M. Christodoulou, A. Riello, and C. Rovelli, Int. J. Mod. Phys. D 21, 1242014 (2012)). Here, we show that the quasiequilibrium thermodynamic state may exist, in which the region with N < 0 has negative local temperature T(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 negative T(r) from the fermions, and thus they may distinguish the spacetimes with positive and negative lapse functions via thermodynamics.