To investigate the evolution in the relation between galaxy stellar and central black hole mass, we identify a population of 508 X-ray-selected active galactic nuclei (AGN) at 0.4 < z < 6 residing within host galaxies with stellar masses in the range 1010 M⊙ < M* < 1012 M⊙. From this sample we construct a volume-limited complete sample of 85 AGN with host galaxy stellar masses M* > 1010.5 M⊙, and specific X-ray luminosities LX > 2.35 × 1043 erg s−1 at 0.4 < z < 3. We calculate the Eddington limiting masses of the supermassive black holes (SMBHs) residing at the centre of these galaxies, and observe an increase in the average Eddington limiting black hole mass with redshift. While the black hole mass and Eddington ratio, μ, are degenerate, if we assume that the local MBH – M* relation holds at all redshifts we find that the mean Eddington ratio μ rises from 0.056 ± 0.010 at z= 0.7 to 0.087 ± 0.011 at z= 1.25, with no significant evolution thereafter to z= 3. Alternatively, by assuming that there is no evolution in μ and then that there is maximum possible evolution to the Eddington limit, we quantify the maximum possible evolution in the M*/MBH ratio as lying in the range 700 < M*/MBH < 10 000, compared with the local value of M*/MBH∼ 1000. We furthermore find that the fraction of galaxies which are AGN (with LX > 2.35 × 1043 erg s−1) rises with redshift from 1.2 ± 0.2 per cent at z= 0.7 to 7.4 ± 2.0 per cent at z= 2.5. We use our results to calculate the maximum time-scales for which our sample of AGN can continue to accrete at their observed rates before surpassing the local galaxy–black hole mass relation. We use these time-scales to calculate the total fraction of massive galaxies which will be active (with LX > 2.35 × 1043 erg s−1) since z= 3, finding that at least ∼40 per cent of all massive galaxies will be Seyfert luminosity AGN or brighter during this epoch. Further, we calculate the energy density due to AGN activity in the Universe as 1.0 (±0.3) × 1057 erg Mpc−3 Gyr−1, potentially providing a significant source of energy for AGN feedback on star formation. We also use this method to compute the evolution in the X-ray luminosity density of AGN with redshift, finding that massive galaxy Seyfert luminosity AGN are the dominant source of X-ray emission in the Universe at z < 3.