Opening Pandora's Box: Maximizing the $q$-entropy with Escort Averages

It is currently a widely used practice to write the constraints in terms of escort averages when the generalized entropies are employed in the maximization scheme. We show that the maximization of the nonadditive $q$-entropy with escort averages leads either to an overall lack of connection with thermodynamics or violation of the second and third laws of thermodynamics if one adopts the Clausius definition of the physical temperature. If an alternative definition of physical temperature is chosen by respecting the divisibility of the total system into independent subsystems, thermodynamic relations are restored albeit at the cost of transforming the nonadditive $q$-entropy into the R\'enyi entropy. These results are illustrated by studying the quantum mechanical free particle.