  In this paper we study the analytic torsion of an odd-dimensional manifold
with isolated conical singularities. First we show that the analytic torsion is
invariant under deformations of the metric which are of higher order near the
singularities. Then we identify the metric anomaly of analytic torsion for a
bounded generalized cone at its regular boundary in terms of spectral
information of the cross-section. In view of previous computations of analytic
torsion on cones, this leads to a detailed geometric identification of the
topological and spectral contributions to analytic torsion, arising from the
conical singularity. The contribution exhibits a torsion-like spectral
invariant of the cross-section of the cone, which we study under scaling of the
metric on the cross-section.
