  Two implicit periodic structures in the solution of sinh-Gordon thermodynamic
Bethe ansatz equation are considered. The analytic structure of the solution as
a function of complex $\theta$ is studied to some extent both analytically and
numerically. The results make a hint how the CFT integrable structures can be
relevant in the sinh-Gordon and staircase models. More motivations are figured
out for subsequent studies of the massless sinh-Gordon (i.e. Liouville) TBA
equation.
