  In this paper we continue the analysis of spectral problems in the setting of
complete manifolds with fibred boundary metrics, also referred to as
$\phi$-metrics, as initiated in our previous work. We consider the Hodge
Laplacian for a $\phi$-metric and construct the corresponding heat kernel as a
polyhomogeneous conormal distribution on an appropriate manifold with corners.
Our discussion is a generalization of an earlier work by Albin and Sher, and
provides a fundamental first step towards analysis of Ray-Singer torsion,
eta-invariants and index theorems in the setting.
