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.
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