  In this work, we study the $1^{--}$ double-gluon charmonium ($\bar{c}ggc$)
and bottomonium ($\bar{b} gg b$) hybrids in terms of QCD sum rules. We find
that the mass of $\bar{c}ggc$ hybrid lies in $M_{H_{c}}$ = $5.33 \sim 5.90$
GeV, while in the bottom sector the mass of $\bar{b}ggb$ hybrid may be situated
in $M_{H_b} = 11.20 \sim 11.68$ GeV. The contributions up to dimension eight at
leading order of $\alpha_s$ (LO) in the operator product expansion are taken
into account in the calculation. The double-gluon charmonium hybrid meson
predicted in this work can decay into a pair of charmed mesons or a pair of
charmed mesons together with a light meson. Especially, we propose to search
for $\bar{c}ggc$ hybrid with $I^G(J^{PC})= 0^-(1^{--})$ in their decay channels
$D \bar{D}/D^* \bar{D}/D^{*} \bar{D}^{*}$ with P wave and $D^* \bar{D}^{*}
\pi/D^* \bar{D}^*\eta/D \bar{D}\rho/D \bar{D}\omega$ with S wave, which may be
accessible in Belle II, PANDA, Super-B, GlueX, and LHCb experiments.
