  We investigate the asymptotic properties of the large deviation function of
the integrated particle current in systems, in or out of thermal equilibrium,
whose dynamics exhibits anomalous diffusion. The physical systems covered by
our study include mutually repelling particles with a drift, a driven lattice
gas displaying a continuous nonequilibrium phase transition, and particles
diffusing in a anisotropic random advective field. It is exemplified how
renormalisation group techniques allow for a systematic determination of power
laws in the corresponding current large deviation functions. We show that the
latter are governed by known universal scaling exponents, specifically, the
anomalous dimension of the noise correlators.
