  We propose a realization of higher-order topological phases in a spring-mass
model with a breathing kagome structure. To demonstrate the existence of the
higher-order topological phases, we characterize the topological properties and
show that the corner states appear under the fixed boundary condition. To
characterize the topological properties, we introduce a formula for the
$\mathbb{Z}_3$ Berry phases in the Brillouin zone. From the numerical result of
this $\mathbb{Z}_3$ Berry phase, we have elucidated that coupling between the
longitudinal and transverse modes yields a state characterized by the Berry
phase $\frac{2\pi}{3}$ for our mechanical breathing kagome model. In addition,
we suggest that the corner states can be detected experimentally through a
forced vibration.
