In a two-mode Gaussian state ˆϱAB, we report on stationary evolution of three measures of correlations defined via the Rényi-2 entropy, i.e. quantum mutual information (QMI) JR,2, the Gaussian–Rényi-2 entanglement (GR2E) ER,2 and Gaussian quantum steering (GQS) G. We evaluate analytical expression of the covariance matrix fully describing the state ˆϱAB. Further, we study, under influences of parameters characterizing the state at hand and its environment, the behavior of the three considered measures. We find that quantum steering G is always upper bounded by (GR2E) ER,2, which in turn is found always upper bounded by half of the QMI JR,2. This therefore satisfies the hierarchical relation 12JR,2≥ER,2≥G established in [L. Lami, C. Hirche, G. Adesso and A. Winter, Phys. Rev. Lett.117 (2016) 220502]. Importantly, we find that both GR2E ER,2 and GQS G are strongly affected by the thermal effects. Remarkably, when the GR2E ER,2 thoroughly vanishes, the GMI JR,2 exhibits a freezing behavior, and seems to be captured within a wide range of temperature.