The pronounced specific heat anomaly observed in cuprate oxide superconductors with Gd layers below the Néel temperature T(subscript N) is studied by using the modified spin-wave theory for an anisotropic layer antiferromagnetic system. The present work elucidates that the anomaly feature is mainly due to the large spin effect of the Gd^(3+)-ion. Our result agrees well with the experiments. In addition, the observed disappearance of the specific heat anomaly in the presence of a magnetic field is also obtained. The specific heat above T(subscript N) is calculated by using a high temperature expansion. The breakdown of the generalized modified spin-wave theory for the large-spin layer antiferromagnetic systems is explicitly demonstrated in the evaluation of the sublattice magnetization and correlation function.