Correlations are essential for the description of the properties of complex systems, particularly from the point of view of statistical mechanics. In this work we discuss the existence of correlations between parts of a non-equilibrium, composite system in a steady state, and its relation with the concept of temperature fluctuations. For this purpose, we review a recently proposed definition of steady-state temperature, namely the fundamental inverse temperature, together with a descriptor of temperature uncertainty, the inverse temperature covariance, showing that both of these quantities are invariants upon the choice of subsystems.