From the definition $\text{Proj} A[X] = \{ \text{ homogeneous ideals }\mathfrak p \in \text{Spec} A[X] \mid (X) \not\subset \mathfrak{p} \}$.
It seems to me that with this definition only the $A$-ideals remain, so that we at least have a homeomorphism on the points. But do we also have $\mathcal O_{\text{Proj} A[X]} = \mathcal O_{\text{Spec} A}$?