We have performed thermal expansion and compressibility measurements on the recently discovered superconducting material NaxCoO2*4xD2O (x=1/3) using neutron powder diffraction over the temperature range 10-295 K and the pressure range 0-0.6 GPa. Pressure measurements were done in a helium-gas pressure cell. Both the thermal expansion and compressibility are very anisotropic, with the largest effects along the c axis, as would be expected for a layered material with weak hydrogen bonding nominally along the c axis. Near room temperature, the anisotropies of the thermal expansion and compressibility of the hexagonal crystal structure are nearly the same [(Dc/c)/(Da/a)=3-4], with a 100 C change in temperature being roughly equivalent to 0.2 GPa pressure. This would imply that changes in atom position parameters are also the same, but this is not the case. While the effects of temperature on the atom positions are essentially what one might expect, the effects of pressure are surprising. With increasing pressure, the thickness of the CoO2 layer increases, due to the combined effects of an increasing Co-O bond length and changes in the O-Co-O angles of the CoO6 octahedra. We conclude that this unusual effect results from pressure-induced strengthening of the hydrogen bonding between the Nax(D2O)4x layers and the CoO2 layers. The strengthening of these hydrogen bonds requires that charge be moved from the somewhere else in the structure; hence, there is a pressure induced charge redistribution that weakens (lengthens) the Co-O bonds and changes the electronic structure of the superconducting CoO2 layers.