  Topological crystalline insulators (TCI) possess electronic states protected
by crystal symmetries, rather than time-reversal symmetry. We show that the
transition metal oxides with heavy transition metals are able to support
nontrivial band topology resulting from mirror symmetry of the lattice. As an
example, we consider pyrochlore oxides of the form A$_2$M$_2$O$_7$. As a
function of spin-orbit coupling strength, we find two $Z_2$ topological
insulator phases can be distinguished from each other by their mirror Chern
numbers, each of which indicates a different TCI. We also derive an effective
$\bf{k\cdot p}$ Hamiltonian, similar to the model introduced for
$\mathrm{Pb_{1-x}Sn_{x}Te}$, and discuss the effect of an on-site Hubbard
interaction on the topological crystalline insulator phase using slave-rotor
mean-field theory, which predicts new classes of topological quantum spin
liquids.
