  We compute the energy dependence of the P_T-integrated cross section of
directly produced quarkonia in pp collisions at next-to-leading order (NLO),
namely up to alpha_s^3, within nonrelativistic QCD (NRQCD). Our analysis is
based on the idea that the P_T-integrated and the P_T-differential cross
sections can be treated as two different observables. The colour-octet NRQCD
parameters needed to predict the P_T-integrated yield can thus be extracted
from the fits of the P_T-differential cross sections at mid and large P_T. For
the first time, the total cross section is evaluated in NRQCD at full NLO
accuracy using the recent NLO fits of the P_T-differential yields at RHIC, the
Tevatron and the LHC. Both the normalisation and the energy dependence of the
J/psi, psi' and Upsilon(1S), we obtained, are in disagreement with the data
irrespective of the fit method. The same is true if one uses CEM-like
colour-octet NRQCD parameters. If, on the contrary, one disregards the
colour-octet contribution, the existing data in the TeV range are well
described by the alpha_s^3 contribution in the colour-singlet model --which, at
alpha_s^4, however shows an unphysical energy dependence. A similar observation
is made for eta(c,b). This calls for a full NNLO or for a resummation of the
initial-state radiation in this channel. In any case, past claims that
colour-octet transitions are dominantly responsible for low-P_T quarkonium
production are not supported by our results. This may impact the interpretation
of quarkonium suppression in high-energy proton-nucleus and nucleus-nucleus
collisions.
