Exercise 11.5 from Atiyah-MacDonald: Hilbert-Serre theorem and Grothendieck group
This is not a true answer, but I think it could be useful: the paper of W. Smoke Dimension and Multiplicity for Graded Algebras treats the case where $A_0$ is a field and obtains a homomorphism $\chi_R$ of $\mathbb{Z}[t]$-modules from $K_{\text{gr}}(A)$ to $\mathbb{Z}[[t]]$. Furthermore, if any finitely generated $A$-graded module have a finite graded free resolution, then $\chi_R$ is an isomorphism from $K_{\text{gr}}(A)$ to $\mathbb{Z}[t]$.