  It has been claimed that the cosmological constant in AdS black holes such as
the BTZ black hole plays the role of the thermodynamic variable of a pressure
in the thermodynamic first law and the Smarr relation from the scaling law of
the Christodoulou-Ruffini formula. However, the dual solution of the BTZ black
hole is the black string which is asymptotically flat despite the presence of
the cosmological constant, and so the explicit form of the pressure with the
role of the cosmological constant is unclear in the black string since the
pressure is subject to the choice of the energy-momentum tensor. Thus, we show
that if the pressure of the black string is still assumed to be proportional to
the cosmological constant similar to the case of the BTZ black hole, then the
thermodynamic first law is consistent with the Smarr relation from the
Christodoulou-Ruffini formula, and the thermodynamic quantities for the
pressure are well-behaved under the dual transformation.
