Thermodynamic relation on black branes with arbitrary cosmological constant

We investigated the Goon-Penco(GP) relationship in (n+1)-dimensional black branes with an arbitrary cosmological constant. Our analysis revealed that the GP relation preserved its form in four-dimensional and (n+1)-dimensional spacetimes, demonstrating its universal behavior with respect to dimensionality. Furthermore, we established that the GP relation exhibits universality across all states of the black hole, including those associated with the event horizon and the cosmological horizon. These findings confirm that the GP relationship remains valid for (n+1)-dimensional black holes and black branes with an arbitrary cosmological constant, independent of the coordinate system employed.