Abstract We report a 72 ks XMM–Newton observation of the Be/X-ray pulsar (BeXRP) RX J0812.4–3114 in quiescence ($L_X ≈ 1.6\times 10^{33}\, \mathrm{erg\, s^{-1}}$). Intriguingly, we find a two-component spectrum, with a hard power-law (Γ ≈ 1.5) and a soft blackbody-like excess below ≈1 keV. The blackbody component is consistent in kT with a prior quiescent Chandra observation reported by Tsygankov et al. and has an inferred blackbody radius of ≈10 km, consistent with emission from the entire neutron star (NS) surface. There is also mild evidence for an absorption line at $≈ 1$ and/or $≈ 1.4\, \mathrm{keV}$. The hard component shows pulsations at P ≈ 31.908 s (pulsed fraction 0.84 ± 0.10), agreeing with the pulse period seen previously in outbursts, but no pulsations were found in the soft excess (pulsed fraction $\lesssim \!31\, {\rm per\, cent}$). We conclude that the pulsed hard component suggests low-level accretion on to the NS poles, while the soft excess seems to originate from the entire NS surface. We speculate that, in quiescence, the source switches between a soft thermal-dominated state (when the propeller effect is at work) and a relatively hard state with low-level accretion, and use the propeller cut-off to estimate the magnetic field of the system to be $\lesssim\! 8.4\times 10^{11}\, \mathrm{G}$. We compare the quiescent thermal LX predicted by the standard deep crustal heating model to our observations and find that RX J0812.4–3114 has a high thermal LX, at or above the prediction for minimum cooling mechanisms. This suggests that RX J0812.4–3114 either contains a relatively low-mass NS with minimum cooling, or that the system may be young enough that the NS has not fully cooled from the supernova explosion.