 Reversible oxygen reduction/evolution reaction (ORR/OER) electrodes with non-precious metal catalysts are essential for the larger scale development of rechargeable metal–air batteries and regenerative fuel cells. Here, an investigation of the catalyst layer morphology is presented with respect to the bifunctional ORR/OER activity and durability of MnO2–LaCoO3 catalyst in 6 M KOH. Graphene, N-doped graphene and multi-walled carbon nanotubes (MWCNT), alone and in combination, were studied as catalyst layer support. The graphene and N-doped graphene microsheets were prepared by ionic liquid-assisted electrochemical exfoliation of graphite. It was found that the hybrid support composed of graphene and MWCNT (1:1 wt) generated up to an order of magnitude higher ORR and OER current densities for MnO2–LaCoO3, compared to either graphene or MWCNT supports individually. The 3D scaffold-like architecture of the graphene microsheets and MWCNT pillars enhances the catalyst layer utilization efficiency and also improves the catalyst anchoring. The latter effects, together with the lower rates of carbon corrosion in the OER region and peroxide generation in the ORR region on the graphene-based supports compared to the reference Vulcan XC-72, contribute to the improved activity of the hybrid catalyst layer as demonstrated using accelerated potential deep cycle experiments.