In order to study structure of proto-neutron stars and those in subsequent cooling stages, it is of great interest to calculate inhomogeneous hot and cold nuclear matter in a variety of phases. The finite-temperature Hartree-Fock-Bogoliubov (FT-HFB) theory is a primary choice for this purpose, however, its numerical calculation for superfluid (superconducting) many-fermion systems in three dimensions requires enormous computational costs. To study a variety of phases in the crust of hot and cold neutron stars, we propose an efficient method to perform the FT-HFB calculation with the three-dimensional (3D) coordinate-space representation. Recently, an efficient method based on the contour integral of Greens function with the shifted conjugate-orthogonal conjugate-gradient method has been proposed [Phys. Rev. C 95, 044302 (2017)]. We extend the method to the finite temperature, using the shifted conjugate-orthogonal conjugate-residual method. We benchmark the 3D coordinate-space solver of the FT-HFB calculation for hot isolated nuclei and fcc phase in the inner crust of neutron stars at finite temperature. The computational performance of the present method is demonstrated. Different critical temperatures of the quadrupole and the octupole deformations are confirmed for $^{146}$Ba. The robustness of the shape coexistence feature in $^{184}$Hg is examined. For the neutron-star crust, the deformed neutron-rich Se nuclei embedded in the sea of superfluid low-density neutrons appear in the fcc phase at the nucleon density of 0.045 fm$^{-3}$ and the temperature of $k_B T=200$ keV. The efficiency of the developed solver is demonstrated for nuclei and inhomogeneous nuclear matter at finite temperature. It may provide a standard tool for nuclear physics, especially for the structure of the hot and cold neutron-star matters.
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