The cohomology of projective spaces over a noetherian ring $A$ is computed in e.g. Hartshorne Chapter III.5. In particular, we know that $H^i(\mathbb{P}_A^n,\mathcal{O}_{\mathbb{P}})$ are finitely generated $A$-modules.
How about these over non-noetherian rings? Do we still have the results analogous to the noetherian cases?
Yes, the same result holds via the same Cech cohomology computation. See Stacks 01XT or EGA III proposition 2.1.12 for a full proof.
The key difference between Hartshorne's approach and that of Stacks/EGA is actually the claim that $H^p(X,\mathcal{F})=0$ for $X$ an affine scheme, $\mathcal{F}$ a quasi-coherent sheaf, and $p>0$. If we knew this, we'd have the desired result by Hartshorne exercise III.4.11 which shows that Cech cohomology is derived functor cohomology on a covering where the higher cohomology of the intersections vanishes (this is a baby case of the Cech-to-derived spectral sequence). Hartshorne only proves the claim that $H^p(X,\mathcal{F})=0$ for $X$ noetherian, but it is true in general (see Stacks 01XB for the precise proof.)