Phase-Space Contractions of Carrollian Black-Hole Thermodynamics

We study Carrollian limits of Schwarzschild-AdS black-hole thermodynamics using covariant phase space. Allowing the cosmological constant to vary, we derive the extended Iyer-Wald identity and identify the renormalized bulk term proportional to $\delta\Lambda$ with the generator-normalized thermodynamic volume contribution $V_\xi\,\delta P$. We show that the Carroll limit contracts the full thermodynamic phase space together with the metric. For fixed Newton constant, the Lorentzian generator $\partial_t$ collapses to a zero-norm direction as $c\to0$, yielding a degenerate sector with vanishing Hamiltonian variation, temperature and volume. Introducing $\xi_\lambda=c^{-\alpha}\partial_t$ and $G=c^\gamma G_C$, we find that the extended first law scales as $c^{1-\alpha-\gamma}$, so finite phase-space contractions require $\alpha+\gamma=1$. The endpoint $(\alpha,\gamma)=(1,0)$, obtained by $\tau=ct$, is the ordinary non-degenerate Lorentzian finite-clock normalization. Carrollian finite first laws lie on the segment $\alpha<1$, hence $\gamma=1-\alpha>0$, and give $T\to0$, $S\to\infty$, with finite $T\,\delta S$ and $V_\xi\,\delta P$. We test the scaling principle for fixed-charge and fixed-rotation AdS black holes, and extend it to arbitrary spacetime dimension within the Schwarzschild-AdS family.