We study the free-electron gas in an Apollonian network within the tight-binding framework. The scale-free and small-world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities, gaps, and minibands. After an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations, we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum. The fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat. The absence of chiral symmetry of the Apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations.