The free energies of the fcc, bcc, hcp, and simple cubic phases for hard spheres are calculated as a function of density using the fundamental measure theory models of Rosenfeld et al. [Phys. Rev. E 55, 4245 (1997)], Tarazona [Phys. Rev. Lett. 84, 694 (2001)], and Roth et al. [J. Phys.: Condens. Matter 14, 12063 (2002)] in the Gaussian approximation. For the fcc phase, the present work confirms the vanishing of the Lindemann parameter (i.e., vanishing of the width of the Gaussians) near close packing for all three models and the results for the hcp phase are nearly identical. For the bcc phase and for packing fractions above eta 0.56, all three theories show multiple solid structures differing in the widths of the Gaussians. In all three cases, one of these structures shows the expected vanishing of the Lindemann parameter at close packing, but this physical structure is only thermodynamically favored over the unphysical structures in the Tarazona theory and even then, some unphysical behavior persists at lower densities. The simple cubic phase is stabilized in the model of Rosenfeld et al. for a range of densities and in the Tarazona model only very near close packing.