We investigate the rotation effect of the RuO$_6$ octahedron around the $c$ axis on the topological and transport properties near the surface of the spin-triplet superconductor Sr$_2$RuO$_4$. While the Fermi level of bulk Sr$_2$RuO$_4$ is near the Lifshitz transition, the RuO$_6$ rotation realized near the surface leads to the change of the Fermi surface topology. The edge current resulting from the time-reversal symmetry breaking in the chiral $p$-wave phase with fully opened excitation gap is less affected around Lifshitz transition. The topological property and the edge state are sensitive to the rotation angle and the amplitude of the nearest neighbor interaction, and the superconducting gap is strongly reduced in the larger next nearest neighbor interaction region. Although the edge state in Sr$_2$RuO$_4$ is topologically protected, it is not robust to the disorder such as impurity or defect.
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