  Let $G$ be a countable discrete amenable group, and $\Lambda$ be a strongly
connected finite $k$-graph. If $(G,\Lambda)$ is a pseudo free and locally
faithful self-similar action which satisfies the finite-state condition, then
the structure of the KMS simplex of the C*-algebra $\O_{G,\Lambda}$ associated
to $(G,\Lambda)$ is described: it is either empty or affinely isomorphic to the
tracial state space of the C*-algebra of the periodicity group
$\Per_{G,\Lambda}$ of $(G,\Lambda)$, depending on whether the Perron-Frobenius
eigenvector of $\Lambda$ preserves the $G$-action. As applications of our main
results, we also exhibit several classes of important examples.
