Methane seems to dissolve more readily in cold water. Why?
The enthalpy of solution going from the ideal gas state to the solution at infinite dilution is exothermic.  However, more importantly, the Gibbs energy of solution is positive meaning that dissolution of methane into water is not a favored process.  Most of this comes from the relatively large negative entropy change associated with the dissolution at room temperature.  This positive Gibbs energy is manifested in the very small solubilities observed and gives rise to the whole concept of something being "hydrophobic."  The fact that any methane dissolves in water is due to equilibrium effects based on a finite positive Gibbs energy.  Since methane is dissolved to a greater extent at lower temperatures must mean that the Gibbs energy is less positive at the lower temperatures.