||Some non-predicted thermal behavior observed in various Ce-lattice compounds close to their critical points are presented. From a comparison between phase diagrams driven by doping (x) and pressure, a pre-critical region (x* < x < x(cr)) is identified in the former group. Some systematic behaviors, observed in the specific heat (C-m/T) and the entropy gain, are recognized as characteristic features of that region. Different measured temperature dependencies of C-m/T are compared, detecting that the onset of the non-Fermi-liquid behavior occurs much closer to x* than x(cr). In a detailed analysis of the evolution of the thermal dependence within the pre-critical region, a further change in the T dependence is frequently observed before to reach the critical point. A generalized power law function is proposed to compare different systems, which allows to identify a low temperature C-m/T(T) anomaly that only involves about 0.01 x R ln 2 of entropy.