  Several thermodynamic measurements for the cobaltate superconductor,
NaxCoO2.yH2O, have so far provided results inconsistent with each other. In
order to solve the discrepancies, we microscopically calculate the temperature
dependences of specific heat and superfluid density for this superconductor. We
show that two distinct specific-heat data from Oeschler et al. and Jin et al.
are reproduced, respectively, for the extended s-wave state and the p-wave
state. Two different superfluid-density data are also reproduced for each case.
These support our recent proposal of possible two different pairing states in
this material. In addition, we discuss the experimentally proposed large
residual Sommerfeld coefficient and extremely huge effective carrier mass.
