Samples fabricated from hot-pressed rods of M0b purity copper processed by the original technology using hydrostatic extrusion and drawing are investigated. The shear modulus are measured in freely damped mode of torsion oscillations, using an automated system of relaxation spectroscopy, which uses the principle of reversible torsion pendulum on a wire sample in the frequency range $f = 48–60$ Hz and the temperature range 293–736 K. The results of measured changes in the elastic properties for two cycles of heating–cooling at the temperature-change rate of 2 K/min during heating are presented. The variations of parameters controlling the contributions to the elastic moduli, based on X-ray diffraction analysis, are investigated. X-ray interference pattern is a superposition of independent $K_{\alpha_{1}}$-, $K_{\alpha_{2}}$-curves. Therefore, at the analysis of the results obtained with the characteristic radiation, a correction for doubling of spectral line is necessary for correct selection of the approximating function and is introduced by the Rechinger method. To reveal the physical nature of the line broadening, a method that allows determining the average size of the coherent scattering regions and mean-square lattice strain by the analysis of the pseudo-Voigt profile for one line is used. Quantitative data of X-ray analysis for the 100 and 111 directions before and after the first heating–cooling cycle are presented for an average size of coherent scattering and, therefore, a size of grains (crystallites), an average strain, and a dislocation density. The experimental results and estimates of contributions to the change in the elastic properties of the three traditional physical mechanisms (elastic moduli of boundaries, the internal stresses, and lattice dislocations) are compared. Under these mechanisms, quantitative estimates of the relative change in the shear modulus before and after the first heating–cooling cycle are made on the basis of X-ray diffraction analysis for 100 and 111 directions. The possibility of using a new two-component model of the single-phase hybrid material for explanation of the anomalous temperature dependence of the elastic moduli of copper with a submicrocrystalline structure is discussed. Data of the volume fractions of the components, which are oriented along 100 and 111 directions, are presented. In particular, they summarize the changes of these volume fractions with time.