We have carried out a survey of long 50 ks XMM–Newton observations of a sample of bright, variable active galactic nuclei (AGN). We found a distinctive energy dependence of the variability in RX J0136.9−3510 where the fractional variability increases from 0.3 to 2 keV, and then remains constant. This is in sharp contrast to other AGN where the X-ray variability is either flat or falling with energy, sometimes with a peak at ∼2 keV superimposed on the overall trend. Intriguingly, these unusual characteristics of the variability are shared by one other AGN, namely RE J1034+396, which is so far unique showing a significant X-ray quasi-periodic oscillation (QPO). In addition, the broad-band spectrum of RX J0136.9−3510 is also remarkably similar to that of RE J1034+396, being dominated by a huge soft excess in the Extreme-UV (EUV) to soft X-ray bandpass. The bolometric luminosity of RX J0136.9−3510 gives an Eddington ratio of about 2.7 for a black hole mass (from the H beta line width) of 7.9 × 107 M⊙. This mass is about a factor of 50 higher than that of RE J1034+396, making any QPO undetectable in this length of observation. None the less, its X-ray spectral and variability similarities suggest that RE J1034+396 is simply the closest representative of a new class of AGN spectra, representing the most extreme mass accretion rates.