The universe is punctured with a high number of black holes that might stand for topological "holes" or "cavities". Starting from this observation, we discuss the possibility to think over the universe in terms of a manifold of very high genus (henceforward VHGM), rather than an isotropic and homogeneous manifold of genus zero. We argue about the feasibility and the likely physical consequences of a cosmic VHGM approach. In ΛCDM, a topological surface of veryhigh genus might guarantee highly stable, topologically ordered structures protected against small perturbations, leadingto fine-tuning and consistency of cosmic parameters. In non-standard cosmological models, a VHGM framework might help to elucidate spatial features like extra-dimensions, compactified universes, spacetime foam, multi-loops stringinteractions. Groups of almost infinite surface diffeomorphisms might also generate topological defects, impurities, passages and handles that connect different regions at either micro-, meso- or macroscopic physical scales, shedding light on the accelerated expansion of the Universe, on wormholes, on multiverse and bubble universes. VHGM might also contribute to the nonlocal interactions that are typical of quantum entanglement and to information and energy storage, with special attention to the quantum vacuum discrepancy and the holographic approaches. In conclusion, the theoreticalpossibility of a cosmic VHGM suggests a strong and methodologically manageable mathematical background subtending the mechanical, dynamical, energetic and informational features of the spacetime fabric in both ΛCDM and non-standardcosmological models.