  We perform a series of 2D smoothed particle hydrodynamics (SPH) simulations
of gas accretion onto binaries via a circumbinary disc, for a range of gas
temperatures and binary mass ratios ($q$). We show that increasing the gas
temperature increases the accretion rate onto the primary for all values of the
binary mass ratio: for example, for $q=0.1$ and a fixed binary separation, an
increase of normalised sound speed by a factor of $5$ (from our "cold" to "hot"
simulations) changes the fraction of the accreted gas that flows on to the
primary from $ 10\%$ to $\sim40\%$. We present a simple parametrisation for the
average accretion rate of each binary component accurate to within a few
percent and argue that this parametrisation (rather than those in the
literature based on warmer simulations) is relevant to supermassive black hole
accretion and all but the widest stellar binaries. We present trajectories for
the growth of $q$ during circumbinary disc accretion and argue that the period
distribution of stellar "twin" binaries is strong evidence for the importance
of circumbinary accretion. We also show that our parametrisation of binary
accretion increases the minimum mass ratio needed for spin alignment of
supermassive black holes to $q \sim 0.4$, with potentially important
implications for the magnitude of velocity kicks acquired during black hole
mergers.
