  On general grounds, a nonequilibrium temperature can be consistently defined
from generalized fluctuation-dissipation relations only if it is independent of
the observable considered. We argue that the dependence on the choice of
observable generically occurs when the phase-space probability distribution is
non-uniform on constant energy shells. We relate quantitatively this observable
dependence to a fundamental characteristics of nonequilibrium systems, namely
the Shannon entropy difference with respect to the equilibrium state with the
same energy. This relation is illustrated on a mean-field model in contact with
two heat baths at different temperatures.
