  We prove an analogue for even dimensional manifolds of the
Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles. We
show that the eta invariant appearing in this result coincides with the eta
invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional
counterpart for manifolds with boundary of the relative index pairing by Lesch,
Moscovici and Pflaum.
