We investigate by molecular simulations thermodynamic properties of hydration water and protein, the sensitivity of hydrogen bonds to change in temperature, and hydration water distribution at varying levels of hydration of a hyperactive antifreeze protein, DAFP-1. Hydration water coverage of the protein and partial thermodynamic properties of the hydration water are heterogeneous, different for the water near the ice-binding site (IBS) and the rest of the protein, particularly at low levels of hydration. Overall, we find the partial specific heat of water to be larger at low hydration levels than in the fully hydrated limit, with the separation corresponding roughly to one hydration layer. Differences in the specific heat in the low- and fully hydrated regions are accounted for by the varying sensitivity of water-water and water-protein hydrogen bonds to change in temperature as a function of hydration, most strikingly near the IBS. Using values computed for the specific heat, we estimate the partial entropy of the water and protein. We find the partial entropy of DAFP-1 to be greater in the fully hydrated limit than at low levels of hydration, whereas the partial entropy of water is somewhat smaller.