  Scalar perturbations around the Kerr black hole in
scalar-Einstein-Gauss-Bonnet (sEGB) theory are studied in the time domain. To
overcome the "outer boundary problem" that usually encountered in traditional
numerical calculations, we apply the hyperboloidal compactification technique
to perform a $(2+1)$-dimensional simulation aiming to obtain a precise object
picture of the wave propagation under the scalar field perturbation. We find
that the big enough coupling constant between the scalar field and the
Gauss-Bonnet curvature is responsible to destroy the original Kerr black hole.
The breakdown of the Kerr spacetime happens earlier and the instability becomes
more violent when the coupling becomes stronger. We further present object
confirmations on the special case for the negative coupling where there exists
a minimum rotation and below which the instability can never happen no matter
how strong the coupling is. We also illustrate the fine structure property in
the quasinormal ringing frequency once there is the coupling, and present the
characteristic imprint of the sEGB theory. We expect that such a fine structure
can be detected in the future gravitational wave observation to test the sEGB
theory.
