  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.
