  We study the KMS states of the C*-algebra of a strongly connected finite
k-graph. We find that there is only one 1-parameter subgroup of the gauge
action that can admit a KMS state. The extreme KMS states for this preferred
dynamics are parameterised by the characters of an abelian group that captures
the periodicity in the infinite-path space of the graph. We deduce that there
is a unique KMS state if and only if the k-graph C*-algebra is simple, giving a
complete answer to a question of Yang. When the k-graph C*-algebra is not
simple, our results reveal a phase change of an unexpected nature in its
Toeplitz extension.
