  Although Boltzmann's definition of entropy, and, hence, the existence of
negative temperatures, are widely accepted, we will show scenarios which
apparently at a first glance are inconsistent with our normal notion of
thermodynamics. This is shown in the framework of stochastic thermodynamics for
special geometrical frustrated systems (GFSs), which have maximum entropy at
its ground state and constant negative temperature. For two GFSs in weak
thermal contact and at same temperature, all energetic constellations are equal
probable. A hot and a cool GFS in contact is driven by entropic forces via heat
transfer to a most probable state in which the hot GFS is in its ground state.
The same holds for a GFS in contact with a gas. As this is not a local maximum
of entropy both subsystems maintain different temperatures here.
Re-parametrization can transform these non-local into local maxima with
corresponding equivalence of re-defined temperatures. However, these
temperatures cannot be assigned solely to a subsystem but describe combinations
of both.The non-local maxima of entropy restrict the naive application of the
zeroth law of thermodynamics. Reformulated this law is still valid with the
consequence that a GFS at constant negative temperature can measure positive
temperatures. A GFS combined with a polarized paramagnetic spin gas may have a
local or non-local maxima of entropy, with equivalent or non-equivalent
temperatures, respectively. In case of a local maximum the spin gas can measure
the temperature of the GFS, however, it does not reveal information about its
energetic state.
