Given a C*-algebra A with a KMS weight for a circle action, we construct and compute a secondary invariant on the equivariant K-theory of the mapping cone of, both in terms of equivariant KK-theory and in terms of a semifinite spectral flow. This in particular puts the previously considered examples of Cuntz algebras [Carey, Phillips, Rennie, Twisted cyclic theory and the modular index theory of Cuntz algebras] and SUq(2) [Carey, Rennie, Tong, J. Geom. Phys. 59: 1431-1452, 2009] in a general framework. As a new example we consider the Araki-Woods IIIλ representations of the Fermion algebra.
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