  It has recently been a common practice to maximize the deformed entropies
through the escort averaging scheme. However, whatever averaging procedure is
employed, one should recover the ordinary Shannon maximization results in the
appropriate limit of the deformation parameter e.g. $q\to 1$ for the Tsallis
and R\'enyi entropies. Otherwise, the very meaning of a consistent
generalization becomes at stake. Using only this equivalence, we show that any
deformed entropy expression, maximized with the escort averaged constraints,
yields that the Shannon entropy $S$ is equal to the logarithm of the ordinary
canonical partition function i.e. $S = \ln(Z_S )$ instead of the correct
thermodynamic relation $S = \beta U + \ln(Z_S )$. Therefore, we conclude that
the use of the escort averaging procedure should be avoided for any deformed
entropies, since it cannot even yield the well-known thermodynamic relations of
the ordinary canonical formalism.
