  We examine the temperature dependence of the optical sum rule in the normal
state due to interactions. To be concrete we adopt a weak coupling approach
which uses an electron-boson exchange model to describe inelastic scattering of
the electrons with a boson, in the Migdal approximation. While a number of
recent works attribute the temperature dependence in the normal state to that
which arises in a Sommerfeld expansion, we show that in a wide parameter regime
this contribution can be quite small. Instead, most of the temperature
dependence arises from the zeroth order term in the `expansion', through the
temperature dependence of the spectral function, and the interaction parameters
contained therein. For low boson frequencies this circumstance causes a linear
T-dependence in the sum rule. We develop some analytical expressions and
understanding of the temperature dependence.
