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 σ to a thermodynamic variable within the extended Iyer–Wald framework. The brane tension acts as a localized vacuum energy with pressure Pσ≡-σ, yielding a new work term VσdPσ 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 Vσ=4π3rh3; for stationary, axisymmetric solutions it admits a covariant, slice-independent definition and evaluates to Vσ=4π3r+3+a2r+. 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 rh/rc. These results establish a consistent flat-braneworld realization of black-hole chemistry in which brane tension provides the physically motivated pressure variable.