Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"?
As $H^*$ is a graded ring, does this question change if we instead look at $Proj(H^*(X))$?
I have so far found a few articles dealing with the spectrum of cohomology rings of varieties, but nothing to do with topological spaces, nor the Proj constriction.
The short answer is yes. For example, $\text{Spec }H^{\bullet}(\mathbb{CP}^{\infty})$ (here the cohomology is concentrated in even degree so there is no issue with applying $\text{Spec}$ in the classical sense) can be identified with the formal affine line. The group structure on $\mathbb{CP}^{\infty}$ given by tensoring line bundles induces a group scheme structure on the formal affine line which is the structure of the formal additive group. This is the beginning of a long story in chromatic homotopy theory connecting homotopy theory with formal groups.
The next example is to replace ordinary cohomology with K-theory, where we get the formal multiplicative group. Now it's natural to wonder if we can find cohomology theories where we get more exotic ($1$-dimensional, commutative) formal groups such as the formal groups attached to elliptic curves, which is part of the story of elliptic cohomology. There are lots of other parts of this story as well; see, for example, Quillen's theorem on MU.
Regarding the complaints in the comments about commutativity, probably a better behaved invariant involves thinking about the derived algebraic geometry of something like $\text{Spec } C^{\bullet}(X)$. DAG XIII explains something like this.
