Let M be a real hypersurface with almost contact metric structure $({phi},{xi},{eta},g)$ of a nonflat complex space form whose structure Jacobi operator $R_{xi}=R({cdot},{xi}){xi}$ is ${xi}$-parallel. In this paper, we prove that the condition ${ abla}_{xi}R_{xi}=0$ characterize the homogeneous real hypersurfaces of type A in a complex projective space $P_n{mathbb{C}}$ or a complex hyperbolic space $H_n{mathbb{C}}$ when $g({ abla}_{xi}{xi},{ abla}_{xi}{xi})$ is constant.