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Based on density functional theory calculations including a Coulomb repulsionparameter $U$, we explore the topological properties of(La$X$O$_3$)$_2$/(LaAlO$_3$)$_4$(111) with $X=$ $4d$ and $5d$ cations. Themetastable ferromagnetic phases of LaTcO$_3$ and LaPtO$_3$ preserve P321symmetry and emerge as Chern insulators (CI) with $C$=2 and 1 and band gaps of41 and 38 meV at the lateral lattice constant of LaAlO$_3$, respectively. Berrycurvatures, spin textures as well as edge states provide additional insightinto the nature of the CI states. While for $X$=Tc the CI phase is furtherstabilized under tensile strain, for $X$=Pd and Pt a site disproportionationtakes place when increasing the lateral lattice constant from $a_{\rm LAO}$ to$a_{\rm LNO}$. The CI phase of $X$=Pt shows a strong dependence on the Hubbard$U$ parameter with sign reversal for higher values associated with the changeof band gap opening mechanism. Parallels to the previously studied($X_2$O$_3$)$_1$/(Al$_2$O$_3$)$_5$(0001) honeycomb corundum layers arediscussed. Additionally, non-magnetic systems with $X$=Mo and W are identifiedas potential candidates for $Z_2$ topological insulators at $a_{\rm LAO}$ withband gaps of 26 and 60 meV, respectively. The computed edge states and $Z_{2}$invariants underpin the non-trivial topological properties.

Chern and $Z_{2}$ topological insulating phases in perovskite-derived $4d$ and $5d$ oxide buckled honeycomb lattices