  Weak-coupling conserving approximations can be constructed by truncations of
the Luttinger-Ward functional and are well known as thermodynamically
consistent approaches which respect macroscopic conservation laws as well as
certain sum rules at zero temperature. These properties can also be shown for
variational approximations that are generated within the framework of the
self-energy-functional theory without a truncation of the diagram series.
Luttinger's sum rule represents an exception. We analyze the conditions under
which the sum rule holds within a non-perturbative conserving approximation.
Numerical examples are given for a simple but non-trivial dynamical two-site
approximation. The validity of the sum rule for finite Hubbard clusters and the
consequences for cluster extensions of the dynamical mean-field theory are
discussed.
