  We propose and investigate the thermodynamic Bethe ansatz equations for the
minimal $W_p^N$ models~(associated with the $A_{N-1}$ Lie algebra) perturbed by
the least~($Z_N$ invariant) primary field $\Phi_N$. Our results reproduce the
expected ultraviolet and infrared regimes. In particular for the positive sign
of the perturbation our equations describe the behaviour of the ground state
flowing from the $W_p^N$ model to the next $W_{p-1}^N$ fixed point.
