  Given a cone pseudodifferential operator $P$ we give a full asymptotic
expansion as $t\to 0^+$ of the trace $\Tr Pe^{-tA}$, where $A$ is an elliptic
cone differential operator for which the resolvent exists on a suitable region
of the complex plane. Our expansion contains $\log t$ and new $(\log t)^2$
terms whose coefficients are given explicitly by means of residue traces. Cone
operators are contained in some natural algebras of pseudodifferential
operators on which unique trace functionals can be defined. As a consequence of
our explicit heat trace expansion, we recover all these trace functionals.
