Co3O4 catalyst derived from Co-MOFs demonstrated rich active sites in the catalytic reaction. In this paper, we prepared the rod-like Co3O4 catalysts by calcining Co-MOFs, which were synthesized by the solvothermal reaction of cobalt nitrate and diverse spatial structure linkers (L3, L4, and L5). Typically, the linkers with different spatial structures (L3, L4, and L5) were synthesized from p-aminobenzoic acid and tetracarboxylic anhydride with different structures (naphthalene-1,4,5,8-tetracarboxylic anhydride, benzene-1,2,4,5-tetracarboxylic anhydride, and perylene-3,4,9,10-tetracarboxylic anhydride). According to the BET analysis, the average pore size of Co3O4 increased with the increase of the linker spatial structure. However, the specific surface area of three Co3O4 increased at first, and then decreased with the increase spatial structure of the linker. Among three Co3O4 catalysts, the Co3O4-L3 exhibited the highest specific surface area. The Co3O4-L3 emerged with superior performance and stability for the catalytic combustion reaction of toluene due to large specific surface area, high redox capacity, plentiful surface active sites, and rich active adsorbed oxygen.