  We calculate entropy-temperature curves for interacting bosons in unit filled
optical lattices for both homogeneous and harmonically trapped situations, and
use them to understand how adiabatic changes in the lattice depth affect the
temperature of the system. In a translationally invariant lattice, the zero
tunneling limit facilitates a rather detailed analytic description. Unlike the
non-interacting bosonic system which is always cooled upon adiabatic loading
for low enough initial temperature, the change in the excitation spectrum
induced by interactions can lead to heating. Finite tunneling helps to reduce
this heating. Finally, we study the spatially inhomogeneous system confined in
a parabolic potential and show that the presence of the trap can significantly
reduce the final available temperature, due to the non-vanishing superfluid
component at the edge of the cloud which is present in trapped systems.
