Thermodynamic relation on rotating charged black strings with arbitrary cosmological constant

The Goon-Penco (GP) relation was investigated on rotating charged black strings with an arbitrary cosmological constant. It has been demonstrated that the GP relation retains its form in the context of spacetimes described by cylindrical coordinates. In addition, the GP relation is derived in scenarios where the energy state parameters (including angular momentum $J$ and charge $Q$, etc.) are expressed as functions of the perturbation parameter $η$. This finding indicates that the GP relation is not only valid for spacetimes described by spherically symmetric coordinates, but also prevalent for non-spherically symmetric spacetimes, such as cylindrical coordinates. Therefore, the present study demonstrated that the GP relation is universal for spacetimes with arbitrary cosmological constants, irrespective of the adopted coordinate system.