  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.
