  We study the finite temperature properties of quantum magnets close to a
continuous quantum phase transition between two distinct valence bond solid
phases in two spatial dimension. Previous work has shown that such a second
order quantum `Lifshitz' transition is described by a free field theory and is
hence tractable, but is nevertheless non-trivial. At $T>0$, we show that while
correlation functions of certain operators exhibit $\omega/T$ scaling, they do
not show analogous scaling in space. In particular, in the scaling limit, all
such correlators are purely {\em local} in space, although the same correlators
at T=0 decay as a power law. This provides a valuable microscopic example of a
certain kind of `local' quantum criticality. The local form of the correlations
arise from the large density of soft modes present near the transition that are
excited by temperature. We calculate exactly the autocorrelation function for
such operators in the scaling limit. Going beyond the scaling limit by
including irrelevant operators leads to finite spatial correlations which are
also obtained.
