  Normal diffusion in corrugated potentials with spatially uncorrelated
Gaussian energy disorder famously explains the origin of non-Arrhenius
$\exp[-\sigma^2/(k_BT^2)]$ temperature-dependence in disordered systems. Here
we show that unbiased diffusion remains asymptotically normal also in the
presence of spatial correlations decaying to zero. However, due to a temporal
lack of self-averaging transient subdiffusion emerges on mesoscale, and it can
readily reach macroscale even for moderately strong disorder fluctuations of
$\sigma\sim 4-5\, k_BT$. Due to its nonergodic origin such subdiffusion
exhibits a large scatter in single trajectory averages. However, at odds with
intuition, it occurs essentially faster than one expects from the normal
diffusion in the absence of correlations. We apply these results to diffusion
of regulatory proteins on DNA molecules and predict that such diffusion should
be anomalous, but much faster than earlier expected on a typical length of
genes for a realistic energy disorder of several room $k_BT$, or merely
$0.05-0.075$ eV.
