  The definition of a nonequilibrium temperature through generalized
fluctuation-dissipation relations relies on the independence of the
fluctuation-dissipation temperature from the observable considered. We argue
that this observable independence is deeply related to the uniformity of the
phase-space probability distribution on the hypersurfaces of constant energy.
This property is shown explicitly on three different stochastic models, where
observable-dependence of the fluctuation-dissipation temperature arises only
when the uniformity of the phase-space distribution is broken. The first model
is an energy transport model on a ring, with biased local transfer rules. In
the second model, defined on a fully connected geometry, energy is exchanged
with two heat baths at different temperatures, breaking the uniformity of the
phase-space distribution. Finally, in the last model, the system is connected
to a zero temperature reservoir, and preserves the uniformity of the
phase-space distribution in the relaxation regime, leading to an
observable-independent temperature.
