Ultrafast heating of solids with modern x-ray free electron lasers (XFELs) leads to a unique set of conditions characterized by the simultaneous presence of heated electrons in a cold ionic lattice. In this work, we analyze the effect of electronic heating on the dynamic structure factor (DSF) in bulk aluminum (Al) with a face-centered cubic lattice and in silicon (Si) with a crystal diamond structure using first-principles linear-response time-dependent density functional theory simulations. We find a thermally induced red shift of the collective plasmon excitation in both materials. In addition, we show that the heating of the electrons in Al can lead to the formation of a double-plasmon peak due to the extension of the Landau damping region to smaller wave numbers. Finally, we demonstrate that thermal effects generate a measurable and distinct signature (peak-valley structure) in the DSF of Si at small frequencies. Our simulations indicate a variety of new features in the spectrum of x-ray-driven solids, specifically at finite momentum transfer, which can be probed in upcoming x-ray Thomson scattering experiments at various XFEL facilities.