I will present a scenario motivated by string dualities and the swampland where the early universe is described by a topological phase of gravity. Many cosmological puzzles can be explained in terms of the topological nature of this phase. To obtain phenomenological predictions, a concrete realization of this scenario using Witten's four dimensional topological gravity will be presented. In this model, the CMB power spectrum is controlled by conformal anomaly coefficients. The model predicts the absence of tensor modes in the CMB fluctuations and significant non-Gaussianities in 4- and higher point functions.
