  We investigated evolution of metric and density perturbations for Tsallis
model of holographic dark energy with energy density $\sim L^{2\gamma - 4}$,
where $L$ is length of event horizon or inverse Hubble parameter and $\gamma$
is parameter of non-additivity close to $1$. Because holographic dark energy is
not an ordinary cosmological fluid but a phenomena caused by boundaries of the
universe, the ordinary analysis for perturbations is not suitable. One needs to
consider perturbations of the future event horizon. For realistic values of
parameters it was discovered that perturbations of dark energy don't grow
infinitely but vanish or freeze. We also considered the case of realistic
interaction between holographic dark energy and matter and showed that in this
case perturbations also can asymptotically freeze with time.
