  $J/\psi$- or $\Upsilon$-pair production at the LHC are promising processes to
study the gluon transverse momentum distributions (TMDs) which remain very
poorly known. In this article, we improve on previous results by including the
TMD evolution in the computation of the observables such as the
pair-transverse-momentum spectrum and asymmetries arising from the linear
polarization of gluons inside unpolarized protons. We show that the azimuthal
asymmetries generated by the gluon polarization are reduced compared to the
tree level case but are still of measurable size (in the 5%-10% range). Such
asymmetries should be measurable in the available data sets of $J/\psi$ pairs
and in the future data sets of the high-luminosity LHC for $\Upsilon$ pairs.
