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

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 the energy-momentum tensor (EMT), which facilitates lattice simulation of the EMT based on the gradient flow. We argue that in the $t to 0$ extrapolation analysis, lattice data should be fitted by a power function in $g(mu(t))$, the flow time dependent running coupling, where the power is determined by the perturbation order we consider. From actual lattice data, we confirm the validity of the extrapolation function. Using the new extrapolation function, we present updated lattice results for thermodynamics quantities in quenched QCD; our results are consistent with the previous study [arXiv:1812.06444] but we obtain smaller errors due to reduction of systematic errors.