  In recent years, Floquet engineering has attracted considerable attention as
a promising approach for tuning topological phase transitions. In this work, we
investigate the effects of high-frequency time-periodic driving in a
four-dimensional (4D) topological insulator, focusing on topological phase
transitions at the off-resonant quasienergy gap. The 4D topological insulator
hosts gapless three-dimensional boundary states characterized by the second
Chern number $C_{2}$. We demonstrate that the second Chern number of 4D
topological insulators can be modulated by tuning the amplitude of
time-periodic driving. This includes transitions from a topological phase with
$C_{2}=\pm3$ to another topological phase with $C_{2}=\pm1$, or to a
topological phase with an even second Chern number $C_{2}=\pm2$ which is absent
in the 4D static system. Finally, the approximation theory in the
high-frequency limit further confirms the numerical conclusions.
