We report a 72 ks XMM-Newton observation of the Be/X-ray pulsar (BeXRP) RX J0812.4-3114 in quiescence ($L_X approx 1.6 times 10^{33}~mathrm{erg~s^{-1}}$). Intriguingly, we find a two component spectrum, with a hard power-law ($Gamma approx 1.5$) and a soft blackbody-like excess below $approx 1~mathrm{keV}$. 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 $approx 10~mathrm{km}$, consistent with emission from the entire neutron star (NS) surface. There is also mild evidence for an absorption line at $approx 1~mathrm{keV}$ and/or $approx 1.4~mathrm{keV}$. The hard component shows pulsations at $P approx 31.908~mathrm{s}$ (pulsed fraction $0.84 pm 0.10$), agreeing with the pulse period seen previously in outbursts, but no pulsations were found in the soft excess (pulsed fraction $lesssim 31%$). We conclude that the pulsed hard component suggests low-level accretion onto the neutron star 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 cutoff to estimate the magnetic field of the system to be $lesssim 8.4 times 10^{11}~mathrm{G}$. We compare the quiescent thermal $L_X$ predicted by the standard deep crustal heating model to our observations and find that RX J0812.4-3114 has a high thermal $L_X$, 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.
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