Is temperature double-counted in the Gibbs free energy equation?
One reason we write equations this way is because all of the parameters on the right-hand-side can be measured or computed independently. 

Consider an analogous (but more intuitive) equation, the Stokes-Einstein relation for the diffusion coefficient $D$ of a particle in a liquid:

$$D = \frac{k_\mathrm{B} T}{6\pi\eta r}$$

where $\eta$ is the dynamic viscosity, $r$ is the radius of the particle, and $k_\mathrm{B}$ is Botzmann's constant. Here we see an explicit dependence of $D$ on $T$, but there is also an implicit dependence on $T$ through $\eta$ and possibly also $r$. Is this a problem? No, assuming the equation holds and we can determine the parameters accurately at each $T$ of interest. Whether that is a simple thing is another question. 

Also, it is fair to wonder to what extent the parameters are fundamentally independent, or, whether by digging into the gory details of atomic structure of matter, we might not find $r$ and $\eta$ to be connected in some other way. But that is another question.