  We theoretically study a multi-band Hubbard model of pyrochlore oxides of the
form A$_2$B$_2$O$_7$, where B is a heavy transition metal ion with strong
spin-orbit coupling, in a thin film geometry orientated along the [111]
direction. Along this direction, the pyrochlore lattice consists of alternating
kagome and triangular lattice planes of B ions. We consider a single kagome
layer, a bilayer, and the two different trilayers. As a function of the
strength of the spin-orbit coupling, the direct and indirect $d$-orbital
hopping, and the band filling, we identify a number of scenarios where a
non-interacting time-reversal invariant Z$_2$ topological phase is expected and
we suggest some candidate materials. We study the interactions in the
half-filled $d$-shell within Hatree-Fock theory and identify parameter regimes
where a zero magnetic field Chern insulator with Chern number $\pm1$ can be
found. The most promising geometries for topological phases appear to be the
bilayer which supports both a Z$_2$ topological insulator and a Chern
insulator, and the triangular-kagome-triangular trilayer which supports a
relatively robust Chern insulator phase.
