The usual sphaleron bound and the statement of the impossibility of baryon production at a second order phase transition or analytic cross-over are reformulated in the first part of the paper as requirements of the expansion rate of the Universe at the electroweak scale. With an (exact or effective) additional contribution to the energy density scaling as 1/a^6, which dominates until just before nucleosynthesis, the observed baryon asymmetry may be produced at the electroweak scale in simple extensions of the Minimal Standard Model, even in the case that the phase transition is not first order. We focus our attention on one such cosmology, in which the Universe goes through a period termed `kination' in which its energy is dominated by the kinetic energy of a scalar field. The required kinetic energy dominated modes can occur either as a field rolls down an exponential (or steeper) potential, or in the oscillation of a field about the minimum of a steep power-law potential. We implement in detail the former case with a single exponential field first driving inflation, and then rolling into a kinetic energy dominated mode. Reheating is achieved using an alternative to the usual mechanism due to Spokoiny, in which the Universe is `reheated' by particle creation in the expanding background. Density perturbations of the magnitude required for structure formation may also be generated. We show that the analogous model for the power-law potential cannot be consistently implemented. In models with inflation driven by a second field and the usual mechanism of reheating (by decay of the inflaton) the required kinetic energy dominated cosmology is viable in both types of potential.