Given a commutative, $\mathbb N$-graded ring, one can associate to it a scheme via the $Proj$ construction.

What happens if one tries to copy this procedure but instead of $\mathbb N$ with another indexing gadget (say commutative monoid) ?

Some thoughts about this: Considering projective varieties is roughly the same as studying affine varieties equivariant under the multiplicative group. So I would guess that replacing $\mathbb N$ by something else corresponds to replacing the multiplicative group by something else.