  The simplest gravity duals for quantum critical theories with z=2 `Lifshitz'
scale invariance admit a marginally relevant deformation. Generic black holes
in the bulk describe the field theory with a dynamically generated momentum
scale Lambda as well as finite temperature T. We describe the thermodynamics of
these black holes in the quantum critical regime where T >> Lambda^2. The
deformation changes the asymptotics of the spacetime mildly and leads to
intricate UV sensitivities of the theory which we control perturbatively in
Lambda^2/T.
