  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
$L_X\gtrsim10^{44}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$, which are consistent with
results for individually detected sources. We also investigated the
$L_X-L_{6\mu\mathrm{m}}$ 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.
