  We study the behaviour of multiple radiative cooling algorithms implemented
in seven Semi-Analytic Models (SAMs) of galaxy formation, including a new model
we propose in this paper. We use versions of the models without feedback and
apply them to dark matter haloes growing in a cosmological context, which have
final masses that range from 10^{11}Msun to 10^{14}Msun. First, using
simplified smoothly-growing halo models, we demonstrate that the different
algorithms predict cooling rates and final cold gas masses that differ by a
factor of ~5 for massive haloes (>10^{12}Msun). The algorithms are in better
agreement for less massive haloes because they cool efficiently and, therefore,
their cooling rates are largely limited by the halo accretion rate. However,
for less massive haloes, all the SAMs predict less cooling than corresponding
1D hydrodynamic models. Second, we study the gas accretion history of the
central galaxies of dark matter haloes using merger trees. The inclusion of
mergers alters the cooling history of haloes by locking up gas in galaxies
within small haloes at early times. For realistic halo models, the dispersion
in the cold gas mass predicted by the algorithms is 0.5 dex for high mass
haloes and 0.1 dex for low mass haloes, while the dispersion in the accretion
rate is about two times larger. Comparing to cosmological SPH simulations, we
find that most SAMs systematically under-predict the gas accretion rates for
low-mass haloes but over-predict the gas accretion rates for massive haloes.
Although the models all include both "rapid" and "slow" mode accretion, the
transition between the two accretion modes varies between models and also
differs from the simulations. Finally, we construct a new model that explicitly
incorporates cold halo gas to illustrate that such a class of models can better
match the results from cosmological hydrodynamic simulations.
