The de Rham-Gabadadze-Tolley (dRGT) black string solution is a cylindrically symmetric and static solution of the Einstein field equation with graviton mass term. For the asymptotically de Sitter (dS) solution, it is possible to obtain the black string with two event horizons corresponding to two thermodynamic systems. The R\'enyi entropy is one of the entropic forms which is suitable to deal with nonextensive properties of the black string. In this work, we investigated the possibility to obtain a stable black string by using the R\'enyi entropy in both separated and effective approaches. We found that the nonextensivity provides the thermodynamically stable black string with moderate size in both approaches. The transition from the hot gas phase to the moderate-sized stable black string in the separated/effective description is a first-order/zeroth-order phase transition. The significant ways to distinguish the black string from both approaches are discussed.

The de Rham-Gabadadze-Tolley (dRGT) black string solution is a cylindrically symmetric and static solution of the Einstein field equation with graviton mass term. For the asymptotically de Sitter (dS) solution, it is possible to obtain the black string with two event horizons corresponding to two thermodynamic systems. The Rényi entropy is one of the entropic forms which is suitable to deal with nonextensive properties of the black string. In this work, we investigated the possibility to obtain a stable black string by using the Rényi entropy in both separated and effective approaches. We found that the nonextensivity provides the thermodynamically stable black string with moderate size in both approaches. The transition from the hot gas phase to the moderate-sized stable black string in the separated/effective description is a first-order/zeroth-order phase transition. The significant ways to distinguish the black string from both approaches are discussed.