  We examine the consistency of the thermodynamics of irrotational and
non-isentropic perfect fluids complying with matter conservation by looking at
the integrability conditions of the Gibbs-Duhem relation. We show that the
latter is always integrable for fluids of the following types: (a) static, (b)
isentropic (admits a barotropic equation of state), (c) the source of a
spacetime for which $r\ge 2$, where $r$ is the dimension of the orbit of the
isometry group. This consistency scheme is tested also in two large classes of
known exact solutions for which $r< 2$, in general: perfect fluid Szekeres
solutions (classes I and II). In none of these cases, the Gibbs-Duhem relation
is integrable, in general, though specific particular cases of Szekeres class
II (all complying with $r<2$) are identified for which the integrability of
this relation can be achieved. We show that Szekeres class I solutions satisfy
the integrability conditions only in two trivial cases, namely the spherically
symmetric limiting case and the Friedman-Roberson-Walker (FRW) cosmology.
Explicit forms of the state variables and equations of state linking them are
given explicitly and discussed in relation to the FRW limits of the solutions.
We show that fixing free parameters in these solutions by a formal
identification with FRW parameters leads, in all cases examined, to unphysical
temperature evolution laws, quite unrelated to those of their FRW limiting
cosmologies.
