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We theoretically clarify the functional form to be used in $t \to 0$extrapolation in the small flow time expansion (SF$t$X) method for theenergy-momentum tensor (EMT), which facilitates lattice simulation of the EMTbased on the gradient flow. We argue that in the $t \to 0$ extrapolationanalysis, lattice data should be fitted by a power function in $g(\mu(t))$, theflow time dependent running coupling, where the power is determined by theperturbation order we consider. From actual lattice data, we confirm thevalidity of the extrapolation function. Using the new extrapolation function,we present updated lattice results for thermodynamics quantities in quenchedQCD; our results are consistent with the previous study [arXiv:1812.06444] butwe obtain smaller errors due to reduction of systematic errors.

$t \to 0$ extrapolation function in SF$t$X method for the energy-momentum tensor