  In the present paper we study the onset of the spin-induced scalarization of
a Kerr black hole in scalar-Gauss-Bonnet gravity with a massive scalar field.
Our approach is based on a (2+1) time evolution of the relevant linearized
scalar field perturbation equation. We examine the region where the Kerr black
hole becomes unstable giving rise to new scalarized rotating black holes with a
massive scalar field. With increasing of the scalar field mass, the minimum
value of the Gauss-Bonnet coupling parameter at which scalarization is
possible, increases and thus the instability region shrinks. Interestingly, the
introduction of scalar field mass does not change the critical minimal value of
the black hole angular momentum $a_{\rm crit}/M$ where the instability of the
Kerr black hole develops.
