BASS LV. Connecting X-ray variability with AGN physical properties and a new path to Cosmological distances

X-ray variability is a well-established characteristic of active galactic nuclei (AGN), known to correlate inversely with both the supermassive black hole mass and luminosity, although the degree of each remains a topic of debate. The potential of X-ray variability as a proxy for MBH or for intrinsic LX has led to proposals to use AGN as standard candles to test cosmological models. However, the large intrinsic dispersion in these relations has limited their practical applications. In this work, we investigate the dependence of X-ray variability on AGN physical properties using a sample of 134 Seyfert 1 galaxies from the BAT AGN Spectroscopic Survey (BASS), which is the largest sample to date, more than three times larger than those used in previous studies. Contrary to earlier findings, we observe that X-ray variability correlates with luminosity just as strongly as with MBH. Furthermore, we still do not find evidence for the expected anti-correlation between variability and Eddington ratio, even when using refined bolometric luminosities from SED fitting to compute the Eddington ratio. From a cosmological perspective, the increased sample size reduces the scatter in the log(L)-log(exvar) relation to ~0.63 dex - a significant improvement over previous results, but still too large to serve as competitive standard candles, when compared to SNIa (uncertainties on distances of ~5-10%) or the L(X)-L(UV) relation in quasars (uncertainties of 10-12%). We tested including the width of broad emission lines as additional parameters, but found that this does not significantly lower the observed dispersion, contrary to previous studies on smaller samples. Finally, we discuss how future X-ray missions such as AXIS and NewAthena will improve this scenario by enabling precise variability measurements for thousands of AGN up to redshift z~3, thereby enabling it as a new cosmological probe.