  When we consider five-dimensional warped Taub-NUT/Bolt AdS black string with
minimally coupled massive scalar field, we calculate entropy by using the brick
wall method. It is found that they are proportional to being quadratically
divergent in a cutoff parameter. In particular, we show that the entropy of
warped Taub-NUT AdS black string holds for an area law in the bulk as well as
on the brane. Furthermore, when the negative cosmological constant is treated
as thermodynamic pressure, we calculate the thermodynamic quantities and
investigate their extended thermodynamic properties. Interestingly, we obtain a
thermodynamically stable range as a function of the temperature for warped
Taub-NUT AdS black string. Finally, we also study a proportional behavior of
the thermodynamic quantities along a warp factor, and find that an entropy, a
specific heat, a Gibbs free energy, and an action difference increase as a warp
factor grows up.
