  We demonstrate that criticality in a driven-dissipative system is strongly
influenced by the spectral properties of the reservoir. We study the
open-system realization of the Dicke model, where a bosonic cavity mode couples
to a large spin formed by two motional modes of an atomic Bose-Einstein
condensate. The cavity mode is driven by a high frequency laser and it decays
to a Markovian bath, while the atomic mode interacts with a colored reservoir.
We reveal that the soft mode fails to describe the characteristics of the
criticality. We calculate the critical exponent of the superradiant phase
transition and identify an inherent relation to the low-frequency spectral
density function of the colored bath. We show that a finite temperature of the
colored reservoir does not modify qualitatively this dependence on the spectral
density function.
