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.
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