  The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for
an integrable spin chain related to the Lie superalgebra osp(1|2) are proposed
by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy
of the QTM and derive the functional relations among them (T-system) and their
certain combinations (Y-system). Their analytical property leads to the
non-linear integral equations which describe the free energy and the
correlation length at any finite temperatures. With regard to the free energy,
they coincide with the TBA equation (math-ph/9911010, Mod. Phys. Lett. A, 14,
2427 (1999)) based on the string hypothesis.
