Hot Dust-Obscured Galaxies (Hot DOGs) are hyperluminous ($L_{mathrm{8-1000,mu m}}>10^{13},mathrm{L_odot}$) infrared galaxies with extremely high (up to hundreds of K) dust temperatures. The sources powering both their extremely high luminosities and dust temperatures are thought to be deeply buried and rapidly accreting supermassive black holes (SMBHs). Hot DOGs could therefore represent a key evolutionary phase in which the SMBH growth peaks. X-ray observations can be used to study their obscuration levels and luminosities. In this work, we present the X-ray properties of the 20 most-luminous ($L_{mathrm{bol}}gtrsim10^{14}, L_odot$) known Hot DOGs at $z=2-4.6$. Five of them are covered by long-exposure ($10-70$ ks) Chandra and XMM-Newton observations, with three being X-ray detected, and we study their individual properties. One of these sources (W0116$-$0505) is a Compton-thick candidate, with column density $N_H=(1.0-1.5)times10^{24},mathrm{cm^{-2}}$ derived from X-ray spectral fitting. The remaining 15 Hot DOGs have been targeted by a Chandra snapshot (3.1 ks) survey. None of these 15 is individually detected; therefore we applied a stacking analysis to investigate their average emission. From hardness-ratio analysis, we constrained the average obscuring column density and intrinsic luminosity to be log$N_H,mathrm{[cm^{-2}]}>23.5$ and $L_Xgtrsim10^{44},mathrm{erg,cm^{-2},s^{-1}}$, which are consistent with results for individually detected sources. We also investigated the $L_X-L_{6mumathrm{m}}$ and $L_X-L_{bol}$ relations, finding hints that Hot DOGs are typically X-ray weaker than expected, although larger samples of luminous obscured QSOs are needed to derive solid conclusions.
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