  Recent theoretical study has shown that the hexagonal YCo$_5$ is dynamically
unstable and distorts into a stable orthorhombic structure. In this study, we
show theoretically that the orthorhombic phase is energetically more stable
than the hexagonal phase in the low-temperature region, while the phonon
entropy stabilizes the hexagonal phase thermodynamically in the
high-temperature region. The orthorhombic-to-hexagonal phase transition
temperature is $\sim$165 K, which is determined using the self-consistent
phonon calculations. We investigate the magnetocrystalline anisotropy energy
(MAE) using the self-consistent and non-self-consistent (force theorem)
calculations with the spin-orbit interaction (SOI) along with the Hubbard $U$
correction. Then, we find that the orthorhombic phase has similar MAE, orbital
moment, and its anisotropy to the hexagonal phase when the self-consistent
calculation with the SOI is performed. Since the orthorhombic phase still gives
magnetic properties comparable to the experiments, the orthorhombic distortion
is potentially realized in the low-temperature region, which awaits
experimental exploration.
