  Black hole spontaneous scalarization has been attracting more and more
attention as it circumvents the well-known no-hair theorems. In this work, we
study the scalarization in Einstein-scalar-Gauss-Bonnet theory with a probe
scalar field in a black hole background with different curvatures. We first
probe the signal of black hole scalarization with positive curvature in
different spacetimes. The scalar field in AdS spacetime could be formed easier
than that in flat case. Then, we investigate the scalar field around AdS black
holes with negative and zero curvatures. Comparing with negative and zero
cases, the scalar field near AdS black hole with positive curvature could be
much easier to emerge. And in negative curvature case, the scalar field is the
most difficult to be bounded near the horizon.
