  We give a classifying theory for $LG$-bundles, where $LG$ is the loop group
of a compact Lie group $G$, and present a calculation for the string class of
the universal $LG$-bundle. We show that this class is in fact an equivariant
cohomology class and give an equivariant differential form representing it. We
then use the caloron correspondence to define (higher) characteristic classes
for $LG$-bundles and to prove for the free loop group an analogue of the result
for characteristic classes for based loop groups in Murray-Vozzo (J. Geom.
Phys., 60(9), 2010). These classes have a natural interpretation in equivariant
cohomology and we give equivariant differential form representatives for the
universal case in all odd dimensions.
