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We study the equation of state of pure SU($2$) gauge theory using Monte Carlosimulations. The scale-setting of lattice parameters has been carried by usingthe gradient flow. We propose a reference scale $t_0$ for the SU($2$) gaugetheory satisfying $t^2 \langle E \rangle|_{t=t_0} =0.1$, which is fixed by anatural scaling-down of the standard $t_0$-scale for the SU($3$) case based onperturbative analyses. We also show the thermodynamic quantities as a functionof $T/T_c$, which are derived by the energy-momentum tensor using the smallflow-time expansion of the gradient flow.

Themodynamics for pure SU($2$) gauge theory using gradient flow