Context. It is well established that between 380 000 and 1 billion years after the Big Bang the Inter Galactic Medium (IGM) underwent a "phase transformation" from cold and fully neutral to warm (≈ 104 K) and ionized. Whether this phase transformation was fully driven and completed by photoionization by young hot stars is a question of topical interest in cosmology. Aims: We propose here that besides the ultraviolet radiation from massive stars, feedback from accreting black holes in high-mass X-ray binaries (BH-HMXBs) was an additional, important source of heating and reionization of the IGM in regions of low gas density at large distances from star-forming galaxies. Methods: We use current theoretical models on the formation and evolution of primitive massive stars of low metallicity, and the observations of compact stellar remnants in the near and distant universe, to infer that a significant fraction of the first generations of massive stars end up as BH-HMXBs. Results: The total number of energetic ionizing photons from an accreting stellar black hole in an HMXB is comparable to the total number of ionizing photons of its progenitor star. However, the X-ray photons emitted by the accreting black hole are capable of producing several secondary ionizations and the ionizing power of the resulting black hole could be greater than that of its progenitor. Feedback by the large populations of BH-HMXBs heats the IGM to temperatures of ≈ 104 K and maintains it ionized on large distance scales. Conclusions: BH-HMXBs determine the early thermal history of the universe and maintain it as ionized over large volumes of space in regions of low density. This has a direct impact on the properties of the faintest galaxies at high redshifts, the smallest dwarf galaxies in the local universe, and on the existing and future surveys at radio wavelengths of atomic hydrogen in the early universe.