  For a Dirac-type operator D with a spectral boundary condition, the
associated heat operator trace has an expansion in powers and log-powers of t.
Some of the log-coefficients vanish in the Atiyah-Patodi-Singer product case.
We here investigate the effect of perturbations of D, by use of a
pseudodifferential parameter-dependent calculus for boundary problems. It is
shown that the first k log-terms are stable under perturbations of D vanishing
to order k at the boundary (and the nonlocal power coefficients behind them are
only locally perturbed). For perturbations of D from the APS product case by
tangential operators commuting with the tangential part A, all the
log-coefficients vanish if the dimension is odd.
