  For massless scalar fields, a relation $\Delta_{n}=\frac{\sqrt{3}}{2}\pi$ for
$n\rightarrow \infty$ was observed in the scalar-Gauss-Bonnet theory. In the
present paper, we extend the discussion by including a nonzero scalar field
mass. For massive scalar fields, we show that the relation
$\Delta_{n}=\frac{\sqrt{3}}{2}\pi$ for $n\rightarrow \infty$ still holds. We
demonstrate this relation with both analytical and numerical methods. The
analytical analysis implies that this relation may be a very universal
behavior.
