Let $K$ be an algebraically closed field. Denote by $\mathcal F_d$ the category of extensions $K\to F$ of transcendent degree $d$.(Objects are field extensions $K\to F$, morphisms are finite extensions $F_1\to F_2$, respecting $K$). Consider the category of functors $Fun(\mathcal F_d, \mathcal{Vect})$ from this category to the category of vector spaces over $\mathbb Q$. What is known about its cohomological dimension?

Especially we have a functor which associate to a field extension $K\to F$ the multiplicative group $(F^\times)\otimes \mathbb Q$. Is it true that all higher cohomology of this functor vanishes?