Caloric and entropic temperatures in non-equilibrium steady states

•Caloric and entropic definitions of nonequilibrium temperature are compared.•Influence of energy flux on temperature of internal variables is studied.•Effects of a heat flux on caloric and entropic temperatures.•Ideal gases and two-level systems are considered.

We examine the non-equilibrium consequences of two different definitions of temperature in systems out of equilibrium: one is based on the internal energy (caloric temperature), and the other one on the entropy (entropic temperature). We discuss the relation between the values obtained from these two definitions in ideal gases and in two-level systems.