  The notion of negative absolute temperature emerges naturally from
Boltzmann's definition of "surface" microcanonical entropy in isolated systems
with a bounded energy density. Recently, the well-posedness of such construct
has been challenged, on account that only the Gibbs "volume" entropy ---and the
strictly positive temperature thereof--- would give rise to a consistent
thermodynamics. Here we present analytical and numerical evidence that
Boltzmann microcanonical entropy provides a consistent thermometry for both
signs of the temperature. In particular, we show that Boltzmann (negative)
temperature allows the description of phase transitions occurring at high
energy densities, at variance with Gibbs temperature.
  Our results apply to nonlinear lattice models standardly employed to describe
the propagation of light in arrays of coupled waveguides and the dynamics of
ultracold gases trapped in optical lattices. Optically induced photonic
lattices, characterized by saturable nonlinearity, are particularly appealing
because they offer the possibility of observing states ---and phase
transitions--- at both signs of the temperature.
