  In this note, we extend the theory of Chern-Cheeger-Simons to construct
canonical invariants for a one-parameter family of flat connections on a smooth
manifold. These invariants lie in degrees $(2p-2)$-cohomology with
$\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in a
variation of paths (parametrising flat connections), in degrees at least three.
