In this talk, I will introduce a particular beyond the Standard Model extension: the 2HDM+a, a non-minimal Higgs sector consisting of two Higgs doublets and a singlet pseudoscalar, a. I will explain how the baryon asymmetry of the Universe can be produced in this model through electroweak baryogenesis and the mechanism of transient CP violation, which naturally avoids electric dipole moment constraints. Theoretical benchmarks created using state-of-the-art methods are then scrutinised with LHC searches and low-energy flavour measurements, leading to viable and predictive scenarios that are testable in the future.