The fixed point algebra $C^*(E)^{gamma}$ of a gauge action $gamma$ on a graph $C^*$-algebra $C^*(E)$ and its AF subalgebras $C^*(E)^{gamma}_{upsilon}$ associated to each vertex v do play an important role for the study of dynamical properties of $C^*(E)$. In this paper, we consider the stability of $C^*(E)^{gamma}$ (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that $C^*(E)^{gamma}$ is stably isomorphic to a graph $C^*$-algebra $C^*(E_{mathbb{Z}};{ imes};E)$ which we observe being stable. We first give an explicit isomorphism from $C^*(E)^{gamma}$ to a full hereditary $C^*$-subalgebra of $C^*(E_{mathbb{N}};{ imes};E)({subset};C^*(E_{mathbb{Z}};{ imes};E))$ and then show that $C^*(E_{mathbb{N}};{ imes};E)$ is stable whenever $C^*(E)^{gamma}$ is so. Thus $C^*(E)^{gamma}$ cannot be stable if $C^*(E_{mathbb{N}};{ imes};E)$ admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue $lambda$ > 1. The AF algebras $C^*(E)^{gamma}_{upsilon}$ are shown to be nonstable whenever E is irreducible. Several examples are discussed.