We find that standard diagrammatic perturbation theory does not exist for the dynamical mean-field theory of the double-exchange model because the vertex function cannot be expanded in terms of the bare vertex function and the full Green's function G (i νl) αα. Nevertheless, a functional Φ satisfying the condition δΦ δG (i νn) αα =Σ (i νn) αα can be constructed because the curl of the self-energy with respect to the Green's function vanishes: δΣ (i νn) αα δG (i νl) ββ -δΣ (i νl) ββ δG (i νn) αα =0. The connection between the functional Φ and the free energy implies that the theory is thermodynamically consistent, meaning that the same thermodynamic properties may be obtained from either the partition function or the Green's function. We provide a concrete example of this consistency by evaluating the magnetic susceptibility and Curie temperature for any Hund's coupling using two such approaches.