  We investigate the linear stability of scalarized black holes (BHs) and
neutron stars (NSs) in the Einstein-scalar-Gauss-Bonnet (GB) theories against
the odd- and even-parity perturbations including the higher multipole modes. We
show that the angular propagation speeds in the even-parity perturbations in
the $\ell \to \infty$ limit, with $\ell$ being the angular multipole moments,
become imaginary and hence scalarized BH solutions suffer from the gradient
instability. We show that such an instability appears irrespective of the
structure of the higher-order terms in the GB coupling function and is caused
purely due to the existence of the leading quadratic term and the boundary
condition that the value of the scalar field vanishes at the spatial
infinity.~This indicates that the gradient instability appears at the point in
the mass-charge diagram where the scalarized branches bifurcate from the
Schwarzschild branch. We also show that scalarized BH solutions realized in a
nonlinear scalarization model also suffer from the gradient instability in the
even-parity perturbations. Our result also suggests the gradient instability of
the exterior solutions of the static and spherically-symmetric scalarized NS
solutions induced by the same GB coupling functions.
