Cosmological consequences of scale-dependent Barrow-Tsallis entropy

We investigate an extended cosmological scenario based on the Barrow-Tsallis entropy, incorporating a varying (i.e., energy-scale-dependent) anomalous dimension. This behavior is reminiscent of quantum gravity and effective field theory settings, where the relevant couplings acquire a nontrivial scale dependence. By applying the gravity-thermodynamic conjecture on the apparent horizon of a flat Friedmann-Robertson-Walker Universe, we derive the corresponding modified cosmological equations. The standard dynamics is recovered as a limiting case when the Barrow-Tsallis entropy reduces to the conventional Bekenstein-Hawking form. The proposed model is tested against a combination of early- and late-time datasets, including observational Hubble parameter measurements, the Pantheon+ catalog of Type~Ia supernovae, the second data release of DESI baryon acoustic oscillations and cosmic microwave background constraints. Using the Bayesian Information Criterion, we finally compare the fitting performance of our framework with that of the $\Lambda$CDM paradigm. While the latter remains mildly favored, our model is shown to be fully compatible with current observations within suitable regions of the parameter space, unveiling a richer phenomenology that points towards a possible alleviation of the Hubble tension.