  In a finite system driven out of equilibrium by a constant external force the
thermodynamic uncertainty relation (TUR) bounds the variance of the conjugate
current variable by the thermodynamic cost of maintaining the non-equilibrium
stationary state. Here we highlight a new facet of the TUR by showing that it
also bounds the time-scale on which a finite system can exhibit anomalous
kinetics. In particular, we demonstrate that the TUR bounds subdiffusion in a
single file confined to a ring as well as a dragged Gaussian polymer chain even
when detailed balance is satisfied. Conversely, the TUR bounds the onset of
superdiffusion in the active comb model. Remarkably, the fluctuations in a comb
model evolving from a steady state behave anomalously as soon as detailed
balance is broken. Our work establishes a link between stochastic
thermodynamics and the field of anomalous dynamics that will fertilize further
investigations of thermodynamic consistency of anomalous diffusion models.
