  Recent discovery of superconductivity in CeRh$_2$As$_2$ clarified an unusual
$H$-$T$ phase diagram with two superconducting phases [Khim et al.
arXiv:2101.09522]. The experimental observation has been interpreted based on
the even-odd parity transition characteristic of locally noncentrosymmetric
superconductors. Indeed, the inversion symmetry is locally broken at the Ce
site, and CeRh$_2$As$_2$ molds a new class of exotic superconductors. The
low-temperature and high-field superconducting phase is a candidate for the
odd-parity pair-density-wave state, suggesting a possibility of topological
superconductivity as spin-triplet superconductors are. In this paper, we first
derive the formula expressing the $\mathbb{Z}_2$ invariant of glide symmetric
and time-reversal symmetry broken superconductors by the number of Fermi
surfaces on a glide invariant line. Next, we conduct a first-principles
calculation for the electronic structure of CeRh$_2$As$_2$. Combining the
results, we show that the field-induced odd-parity superconducting phase of
CeRh$_2$As$_2$ is a platform of topological crystalline superconductivity
protected by the nonsymmorphic glide symmetry and accompanied by boundary
Majorana fermions.
