Numerical-relativity simulation is performed for rapidly spinning black holes (BHs) in a higher-dimensional spacetime of special symmetries for the dimensionality 6 <= d <= 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 [M. Shibata and H. Yoshino, Phys. Rev. D 81, 021501(R) (2010).]. The critical values of a nondimensional spin parameter for the onset of the instability are q := a/mu(1/(d-3)) approximate to 0.74 for d = 6, approximate to 0.73 for d = 7, and approximate to 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(crit)) appear to be dynamically stable for any perturbation. Long-term 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(crit). For more rapidly spinning unstable BHs, the time scale, 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(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.