  (R-channel) TBA is elaborated to find the effective central charge dependence
on the boundary parameters for the massless boundary sine-Gordon model with the
coupling constant $(8\pi) /\beta^2 = 1+ \lambda $ with $\lambda$ a positive
integer. Numerical analysis of the massless boundary TBA demonstrates that at
an appropriate boundary parameter range (cusp point) there exists a singularity
crossing phenomena and this effect should be included in TBA to have the right
behavior of the effective central charge.
