  Numerical-relativity simulation is performed for rapidly spinning black holes
(BHs) in a higher-dimensional spacetime of special symmetries for the
dimensionality $6 \leq d \leq 8$. We find that higher-dimensional BHs, spinning
rapidly enough, are dynamically unstable against nonaxisymmetric bar-mode
deformation and spontaneously emit gravitational waves, irrespective of $d$ as
in the case $d=5$ \cite{SY09}. The critical values of a nondimensional spin
parameter for the onset of the instability are $q:=a/\mu^{1/(d-3)} \approx
0.74$ for $d=6$, $\approx 0.73$ for $d=7$, and $\approx 0.77$ for $d=8$ where
$\mu$ and $a$ are mass and spin parameters. Black holes with a spin smaller
than these critical values ($q_{\rm crit}$) appear to be dynamically stable for
any perturbation. Longterm simulations for the unstable BHs are also performed
for $d=6$ and 7. We find that they spin down as a result of gravitational-wave
emission and subsequently settle to a stable stationary BH of a spin smaller
than $q_{\rm crit}$. For more rapidly spinning unstable BHs, the timescale, for
which the new state is reached, is shorter and fraction of the spin-down is
larger. Our findings imply that a highly rapidly spinning BH with $q > q_{\rm
crit}$ cannot be a stationary product in the particle accelerators, even if it
would be formed as a consequence of a TeV-gravity hypothesis. Its implications
for the phenomenology of a mini BH are discussed.
