I have no experience with the theory of support varieties and fg-conditions beside reading some introductiory articles. I use the survey article “Support varieties for modules and complexes” by Oyvind Solberg [S] here as a reference for definitions and results, see https://folk.ntnu.no/oyvinso/Papers/workshopAMS.ps . Let $ A$ be a finite dimensional algebra with Jacobson radical $ J$ . $ A$ is said to satisfy the Fg1 and Fg2 conditions if and only if the Hochschild cohomology ring $ HH^{*}(A)$ of A is noetherian and $ Ext_A^{*}(A/J,A/J)$ is a finitely generated module over $ HH^{*}(A)$ , see proposition 5.7. in [S]. Now in case $ A$ satifies the Fg1 and Fg2 conditions, it should be true that every indecomposable module of complexity equal to one should be a periodic module (see theorem 5.9. (c) and 7.1. (b) in [S]). Furthermore, satisfying Fg1 and Fg2 implies that $ A$ is a Gorenstein algebra, see theorem 5.9. (a) in [S].

Now here comes where I probably have a thinking error:

Let $ A$ be of finite Gorenstein dimension $ g>0$ with infinite globlal dimension. Assume $ A$ has only finitely many Gorenstein projective modules (a module is Gorenstein projective iff it is in $ \Omega^g(mod-A)$ ). Then a module $ M$ is Gorenstein projective indecomposable iff it is periodic. I would think that this implies that $ A$ can never satisfy the Fg1 and Fg2 conditions because there should exist an indecomposable module $ M$ that is not Gorenstein projective with infinite projective dimension. Then $ \Omega^g(M)$ is Gorenstein projective and thus periodic. This implies that $ M$ has complexity one but is not periodic and thus the Fg1 and Fg2 conditions can not hold?

Let me give a concrete example:

Let $ A$ be the Nakayama algebra with Kupisch series [4,5] given by quiver and relations (points numbered from 0 to 1). This algebra is 2-Gorenstein and of course has only finitely many indecomposable Gorenstein projectives since it is representation-finite.

Let $ S_0$ be the simple module corresponding to the point 0. Then $ S_0$ is not Gorenstein projective but the module $ \Omega^2(S_0)=e_0 A/ e_0 J^2$ is Gorenstein projective and also periodic. This gives that $ S_0$ has complexity equal to one but is not periodic. Thus $ A$ should not satisfy the Fg1 and Fg2 conditions. Likewise I think any Nakayama algebra with infinite global dimension that is not selfinjective can not satisfy the Fg1 and Fg2 conditions in case my argument makes sense.

On the other hand the article http://fuji.cec.yamanashi.ac.jp/ring/oldmeeting/2010/report2010/Nagase.pdf claims that all Nakayama algebras that are Gorenstein satisfy Fg1 and Fg2 so I am confused now.