  We study the linear stability of spontaneously scalarized black holes (BHs)
induced by a scalar field $\phi$ coupled to a Gauss-Bonnet (GB) invariant
$R_{\rm GB}^2$. For the scalar-GB coupling $\xi(\phi)=(\eta/8) (\phi^2+\alpha
\phi^4)$, where $\eta$ and $\alpha$ are constants, we first show that there are
no angular Laplacian instabilities of even-parity perturbations far away from
the horizon for large multipoles $l \gg 1$. The deviation of angular
propagation speeds from the speed of light is largest on the horizon, whose
property can be used to put constraints on the model parameters. For $\alpha
\gtrsim -1$, the region in which the scalarized BH is subject to angular
Laplacian instabilities can emerge. Provided that $\alpha \lesssim -1$ and
$-1/2<\alpha \phi_0^2<-0.1155$, where $\phi_0$ is the field value on the
horizon with a unit of the reduced Planck mass $M_{\rm Pl}=1$, there are
scalarized BH solutions satisfying all the linear stability conditions
throughout the horizon exterior. We also study the stability of spontaneously
scalarized BHs in scalar-GB theories with a nonminimal coupling $-\beta \phi^2
R/16$, where $\beta$ is a positive constant and $R$ is a Ricci scalar. As the
amplitude of the field on the horizon approaches an upper limit
$|\phi_0|=4/\sqrt{\beta}$, one of the squared angular propagation speeds
$c_{\Omega-}^2$ enters the instability region $c_{\Omega-}^2<0$. So long as
$|\phi_0|$ is smaller than a maximum value determined for each $\beta$ in the
range $\beta>5$, however, the scalarized BHs are linearly stable in both
angular and radial directions.
