Since the discovery of high superconducting transition temperature in layered copper oxides, many researchers have searched for similar behavior in other layered metal oxides, such as cobalt and nickel. The sodium cobalt oxyhydrate is the first cobalt-oxide layered superconductor. We present the studies of low-temperature specific heat C(T, H) in NaxCoO2¡EyH2O (x ~ 0.35, y ~ 1.3). At H = 0, a very sharp anomaly was observed at T ~ 4.7 K indicating the existence of bulk superconductivity. There exists an £\T2 term in C(T, H=0) in the superconducting state manifesting the line nodal superconducting order parameter. The feature at the superconducting transition is rather sharp, becoming broad and strongly suppressed in an applied magnetic field. The transition temperature also changed in an applied magnetic field. Thus an abrupt change of slope in H vs. Tc curve was observed. Possible scenarios such as the multiple phase transitions in the mixed state are discussed.