If there were such a functional $\mathcal{F}$, observe that
Under Ricci flow the functional would have to decrease. That is, if $\partial_t g(t) = -2 Rc[g]$ then $\partial_t \mathcal{F}(g(t)) \leq 0$, with strict inequality of $-2 Rc[g] \neq 0$.
The functional would have to be invariant under diffeomorphisms, that is if $\Phi$ is a diffeomorphism of the manifold then $\mathcal{F}(\Phi^*g) = \mathcal{F}(g)$. This is because the Ricci flow is invariant under diffeomorphism.
The point is that Robert Bryant found a metric on $\mathbb R^n$ which moves under Ricci flow only by diffeomorphisms (which is called a steady soliton). That is, there is a metric $g_S$ and a family of diffeomorphisms $\Phi_t$ such that $g(t) = \Phi^*_t g_S$ is the solution of Ricci flow starting from $g_S$. By point 2 above, $\mathcal{F}(g(t)) = \mathcal{F}(g_S)$, which contradicts point 1.
The details of the constructing the soliton are written by Robert Bryant here. Many other gradient steady solitons have been discovered since then.