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Recently, Harlander et al.\ [Eur.\ Phys.\ J.\ C {\bf 78}, 944 (2018)] havecomputed the two-loop order (i.e., NNLO) coefficients in the gradient-flowrepresentation of the energy--momentum tensor (EMT) in vector-like gaugetheories. In this paper, we study the effect of the two-loop order corrections(and the three-loop order correction for the trace part of the EMT, which isavailable through the trace anomaly) on the lattice computation ofthermodynamic quantities in quenched QCD. The use of the two-loop ordercoefficients generally reduces the $t$~dependence of the expectation values ofthe EMT in the gradient-flow representation, where $t$~is the flow time. Withthe use of the two-loop order coefficients, therefore, the $t\to0$extrapolation becomes less sensitive to the fit function, the fit range, andthe choice of the renormalization scale; the systematic error associated withthese factors is considerably reduced.

Thermodynamics in quenched QCD: energy--momentum tensor with two-loop order coefficients in the gradient flow formalism