  In this paper, we introduce and study the annealed spectral sample of Voronoi
percolation, which is a continuous and finite point process in $\mathbb{R}^2$
whose definition is mostly inspired by the spectral sample of Bernoulli
percolation introduced in [GPS10] by Garban, Pete and Schramm. We show a
clustering effect as well as estimates on the full lower tail of this spectral
object. Our main motivation is the study of two models of dynamical critical
Voronoi percolation in the plane. In the first model, the Voronoi tiling does
not evolve in time while the colors of the cells are resampled at rate $1$. In
the second model, the centers of the cells move according to (independent) long
range stable L\'evy processes but the colors do not evolve in time. We prove
that for these two dynamical processes there exist almost surely exceptional
times with an unbounded monochromatic component.
