The standard double cover of a graph $Γ$ is the direct product $Γ\times K_2$. A graph $Γ$ is said to be stable if all the automorphisms of $Γ\times K_2$ come from its factors. Although the study of stability has attracted significant attention, particularly regarding Cayley graphs of abelian groups, a complete classification remains elusive even for Cayley graphs of cyclic groups. In this paper, we study the asymptotic enumeration of both labeled and unlabeled Cayley graphs of abelian groups whose standard double cover has the smallest possible automorphism group. As a corollary, in both the labeled and unlabeled settings, we conclude that the proportion of stable Cayley graphs of an abelian group of order $r$ approaches $1$ as $r\rightarrow\infty$, proving that almost all Cayley graphs of finite abelian groups are stable.