Extended black hole thermodynamics in a DGP braneworld

We develop extended black-hole thermodynamics on a Dvali–Gabadadze–Porrati (DGP) brane by promoting the brane tension $\sigma$ to a thermodynamic variable within the extended Iyer–Wald framework. The brane tension acts as a localized vacuum energy with pressure $P_{\sigma }\equiv -\sigma$, yielding a new work term $V_{\sigma }\mathrm{d}{P}_{\sigma }$ in the first law and the corresponding Smarr relation. For static, spherically symmetric black holes we show that the conjugate volume equals the geometric volume $V_{\sigma }=\frac{4\pi }{3}{r}_h^3$; for stationary, axisymmetric solutions it admits a covariant, slice-independent definition and evaluates to $V_{\sigma }=\frac{4\pi }{3}\text{\hspace{0.17em}⁣}(r_{+}^3+a^2{r}_{+})$. Working on the ghost-free normal branch, the brane is asymptotically flat with a single horizon, so the construction avoids de Sitter obstructions. Along a flat-brane path, asymptotic flatness is preserved by co-varying the bulk cosmological constant, and induced-gravity effects are suppressed by $r_h\mathrm{/}{r}_c$. These results establish a consistent flat-braneworld realization of black-hole chemistry in which brane tension provides the physically motivated pressure variable.