Phantom-Divide Crossing in Barrow-Tsallis Holographic Dark Energy with a Scale-Dependent Barrow Exponent
Phantom-divide crossing is studied in spatially flat cosmologies with standard Friedmann equations, separately conserved holographic dark energy and a barotropic fluid, and the future event horizon as the infrared cutoff. The horizon entropy and dark-energy density are prescribed by $ S_h=S_{\mathrm{BH}}F(L) $ and $ ρ_X\propto F(L)/L^2 $, where $ F(L) $ is smooth, positive and single-valued, with $ χ=\mathrm{d}\ln F/\mathrm{d}\ln L $. For $ w_b>-1 $, $ 0<Ω_X<1 $ and $ χ<2 $ along the trajectory, at most one phantom-divide crossing is possible. If a crossing occurs, dark energy passes from the quintessence regime into the phantom regime at a maximum of the horizon radius. The crossing kinematics determines the local entropy-scaling dimension and the sign of the local Chevallier-Polarski-Linder coefficient. Integrating the locally prescribed scale-dependent Barrow exponent reconstructs the Barrow-Tsallis entropy function. A necessary and sufficient event-horizon consistency criterion selects admissible solutions, whose late-time asymptotics include de Sitter states, Type III and Big Rip singularities, and accelerated power-law expansion under the corresponding parameter conditions. A numerical comparison with a constant Barrow exponent distinguishes the occurrence of a crossing from the effect of exponent variation on late-time evolution. With an adiabatic barotropic fluid, additive total entropy and an equilibrium Gibbs relation for dark energy, the generalized second law fails at the kinematic crossing for positive barotropic-fluid entropy and finite nonzero dark-energy temperature. The divergence of $ c_a^2 $ shows that the purely barotropic adiabatic description of perturbations breaks down. This divergence alone does not establish physical instability.