What are the degrees of freedom (from Gibbs phase rule) for an azeotropic mixture?
Gibbs' phase rule simply counts the number of variables required to specify the system and the number of relationships between these variables (constraints). Any failure of Gibbs' phase rule is symptomatic of additional variables or constraints that have not been taken into account.

This system is generally specified by four variables, the mole fraction of one species in the liquid phase, $x$, the same in the gas phase, $y$, the pressure $P$, and the temperature $T$. Equilibrium of the liquid and gas phases in both species provides two constraints. The specification of an azeotrope, for which $x = y$, provides a third constraint, as has been mentioned in the comments. One degree of freedom remains.