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 \(V_\sigma \,\textrm{d}P_\sigma \) 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 \(V_\sigma =\tfrac{4\pi }{3}r_h^3\) ; for stationary, axisymmetric solutions it admits a covariant, slice-independent definition and evaluates to \(V_\sigma =\tfrac{4\pi }{3}\!\left( r_+^3+a^2 r_+\right) \) . 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 \(r_h/r_c\) . These results establish a consistent flat-braneworld realization of black-hole chemistry in which brane tension provides the physically motivated pressure variable.
