We report a study of aqueous solutions of poly(vinylalcohol) and its hydrogel by thermal conductivity,κ, and specific heat measurements. In particular, we investigate (i) the changes in the solution and the hydrogel at 0.1 MPa observed in the 350-90 K range and of the frozen hydrogel at 130 K observed in the range from 0.1 MPa to 1.3 GPa, and (ii) the nature of the pressure collapse of ice in the frozen hydrogel and kinetic unfreezing on heating of its high density water at 1 GPa. The water component of the polymer solution on cooling either first phase separates and then freezes to hexagonal ice or freezes without phase separation and the dispersed polymer chains freeze-concentrate in nanoscopic and microscopic regions of the grain boundaries and grain junctions of the ice crystals in the frozen state of water in the hydrogel. The change in κ with temperature at 1 bar is reversible with some hysteresis, but not reversible with pressure after compression to 0.8 GPa at 130 K. At high pressures the crystallized state collapses showing features of. and specific heat characteristic of formation of high density amorphous solid water. The pressure of structural collapse is 0.08 GPa higher than that of ice at 130 K. The slowly formed collapsed state shows kinetic unfreezing or glass-liquid transition temperature at 140 K for a time scale of 1 s. Comparison with the change in the properties observed for ice shows that κ decreases when the polymer is added.