We compute the energy dependence of the P Tintegrated cross section of directly produced quarkonia in pp collisions at next-to-leading order (NLO), namely up to α 3 S , within nonrelativistic QCD (NRQCD), treating the P T -integrated and the P T -differential cross sections 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. Both the normalisation and the energy dependence of the J/ψ, ψ and ϒ(1S) we obtained disagree with the data except when using the fit results of Butenschoen and Kniehl. If one disregards the colour-octet contribution, the existing data in the TeV range are well described by the α 3 S contribution in the coloursinglet model -which, at α 4 S , however, shows an unphysical energy dependence. A similar observation is made for η c,b . All this underlines the necessity for a resummation of initial-state radiations in both channels, which is, however, beyond the scope of this article. In any case, past claims that colour-octet transitions are dominantly responsible for low-P T quarkonium production are not supported by our results.