A central aspect of the cosmological constant problem is to understand why vacuum energy does not gravitate. In order to account for this observation, while allowing for nontrivial dynamics of the quantum vacuum, we motivate a novel background independent theory of gravity. The theory is an extension of unimodular gravity that is described in geometric terms by means of a conformal (light-cone) structure and differential forms of degree one and two. We show that the subset of the classical field equations describing the dynamics of matter degrees of freedom and the conformal structure of spacetime are equivalent to that of unimodular gravity. The sector with vanishing matter fields and flat conformal structure is governed by the field equations of BF theory and contains topological invariants that are influenced by quantum vacuum fluctuations. Perturbative deviations from this sector lead to classical solutions that necessarily display relatively small values of the cosmological constant with respect to the would-be contribution of quantum vacuum fluctuations. This feature that goes beyond general relativity (and unimodular gravity) offers an interpretation of the smallness of the currently observed cosmological constant.

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