This paper reported the fabrication and performance of shape-stable composite phase change materials (PCMs) based on lauric acid (LA) and graphene/graphene oxide (GO) complex aerogels for enhancement of thermal energy storage and electrical conduction. The graphene/GO complex aerogels were prepared through a reduction reaction and freeze-drying technology, and then LA was incorporated into the complex aerogels via vacuum-assisted impregnation. The complex aerogels and their composites were characterized by scanning electron microscopy, X-ray powder diffraction and Fourier-transform infrared spectroscopy. These LA/complex aerogel composites not only exhibited high electrical conductivity but also achieved high phase-change enthalpies more than 198.1/g, high heat-charging and discharging efficiency more than 90%, excellent cyclic stability, good phase-change reversibility and good shape stability. The crystallization kinetic study demonstrated that the aerogel framework could generate a heterogeneous nucleation effect on the crystallization of LA and therefore improved the crystallization rate of LA under both nonisothermal and isothermal conditions.