  One of the main predictions of excursion set theory is that the clustering of
dark matter haloes only depends on halo mass. However, it has been long
established that the clustering of haloes also depends on other properties,
including formation time, concentration, and spin; this effect is commonly
known as halo assembly bias. We use a suite of gravity-only simulations to
study the dependence of halo assembly bias on cosmology; these simulations
cover cosmological parameters spanning 10$\sigma$ around state-of-the-art
best-fitting values, including standard extensions of the $\Lambda$CDM paradigm
such as neutrino mass and dynamical dark energy. We find that the strength of
halo assembly bias presents variations smaller than 0.05 dex across all
cosmologies studied for concentration and spin selected haloes, letting us
conclude that the dependence of halo assembly bias upon cosmology is
negligible. We then study the dependence of galaxy assembly bias (i.e. the
manifestation of halo assembly bias in galaxy clustering) on cosmology using
subhalo abundance matching. We find that galaxy assembly bias also presents
very small dependence upon cosmology ($\sim$ 2$\%$-4$\%$ of the total
clustering); on the other hand, we find that the dependence of this signal on
the galaxy formation parameters of our galaxy model is much stronger. Taken
together, these results let us conclude that the dependence of halo and galaxy
assembly bias on cosmology is practically negligible.
