We studied the energy barrier for proton conduction with volume variation in $BaZrO_3$ using a first principles study to investigate an optimum volume for the proton conduction. The volume increase of $BaZrO_3$ was expected to decrease the energy barrier for proton rotation and to increase that for proton transfer, and these trends could be extrapolated when the volume was decreased. However, the energy barriers for the proton transfer with the volume decrease were increased, while all the other energy barriers varied as expected. We could explain this unexpected behavior by the bent Zr-O-Zr structure, when the volume was decreased.