  The tachyonic instability of the Kerr black holes is analyzed in the
Einstein-scalar theory with the quadratic scalar couplings to two topological
terms which are parity-even Gauss-Bonnet and parity-odd Chern-Simons terms. For
positive coupling $\alpha$, we use the (2+1)-dimensional hyperboloidal
foliation method to derive the threshold curve which is the boundary between
stable and unstable Kerr black holes by considering a spherically symmetric
scalar-mode perturbation. In case of negative coupling, a newly bound of
$a\ge0.26$ with $a$ the rotation parameter is found for unstable Kerr black
holes and its threshold curve is derived.
