  Both the Barrow and Tsallis $\delta$ entropies are one-parameter
generalizations of the black-hole entropy, with the same microcanonical
functional form. The ensuing deformation is quantified by a dimensionless
parameter $\Delta$, which in the case of Barrow entropy represents the
anomalous dimension, while in Tsallis' case, it describes the deviation of the
holographic scaling from extensivity. Here, we utilize the
gravity-thermodynamics conjecture with the Barrow--Tsallis entropy to
investigate the implications of the related modified Friedmann equations on the
spectrum of primordial gravitational waves. We show that, with the experimental
sensitivity of the next generation of gravitational wave detectors, such as the
Big Bang Observer, it will be possible to discriminate deviations from the
$\Lambda$CDM model up to $\Delta\lesssim\mathcal{O}(10^{-3})$.
