The graphene oxide (GO) membrane treated by ozone, GOL, exhibits a relatively high conductivity, but its mechanical strength cannot be maintained due to severe swelling. To further improve the conductivity while suppressing the swelling, the free-standing GO membrane was subjected to ozone treatment and subsequently modified by crosslinking agents with oxygen functional groups, i.e., glyoxal (GLX) and oxalic acid (OXA). The modified membranes, GLX/GOL and OXA/GOL, successfully reduced the swelling and increased the conductivity to provide an excellent power output of a fuel cell at 30 °C due to the increased oxygen functional groups and the carbon defects. Furthermore, OXA/GOL exhibited a significant stability at elevated temperatures. At 50 °C, the through-plane conductivity and power output increased to 23.5 mS cm−1 and 317 mW cm−2, respectively. This is considered to be due to the relatively high thermal stability of the carboxy group added by the OXA modification.