  Among observables characterising the random exploration of a graph or
lattice, the cover time, namely the time to visit every site, continues to
attract widespread interest. Much insight about cover times is gained by
mapping to the (spaceless) coupon-collector problem, which amounts to ignoring
spatio-temporal correlations, and an early conjecture that the limiting cover
time distribution of regular random walks on large lattices converges to the
Gumbel distribution in $d \geq 3$ was recently proved rigorously. Furthermore,
a number of mathematical and numerical studies point to the robustness of the
Gumbel universality to modifications of the \textit{spatial} features of the
random search processes (e.g.\ introducing persistence and/or intermittence, or
changing the graph topology). Here we investigate the robustness of the Gumbel
universality to dynamical modification of the \textit{temporal} features of the
search, specifically by allowing the random walker to "accelerate" or
"decelerate" upon visiting a previously unexplored site. We generalise the
mapping mentioned above by relating the statistics of cover times to the
roughness of $1/f^\alpha$ Gaussian signals, leading to the conjecture that the
Gumbel distribution is but one of a family of cover time distributions, ranging
from Gaussian for highly accelerated cover, to exponential for highly
decelerated cover. While our conjecture is confirmed by systematic Monte Carlo
simulations in dimensions $d > 3$, our results for acceleration in $d=3$
challenge the current understanding of the role of correlations in the cover
time problem.
