  The observation of negative-temperature states in the localized phase of the
the Discrete Nonlinear Schr\"odinger (DNLS) equation has challenged statistical
mechanics for a long time. For isolated systems, they can emerge as stationary
extended states through a large-deviation mechanism occurring for finite sizes,
while they are formally unstable in grand-canonical setups, being associated to
an unlimited growth of the condensed fraction. Here, we show that
negative-temperature states in open setups are metastable but their lifetime
$\tau$ is exponentially long with the temperature, $\tau \approx \exp(\lambda
|T|)$ (for $T<0$ and $\lambda>0$). More precisely, we find that condensation on
a given site (i.e. emergence of a tall discrete breather) can advance only once
a critical mass has been accumulated therein. This result has been obtained by
combining the development of an effective grand-canonical formalism with the
implementation of suitable heat baths. A general expression for $\lambda$ is
obtained in the case of a simplified stochastic model of non-interacting
particles. In the DNLS model, the presence of an adiabatic invariant, makes
$\lambda$ even larger because of the resulting freezing of the breather
dynamics. This mechanism, based on the existence of two conservation laws,
provides a new perspective over the statistical description of condensation
processes.
