  We analyze bulk thermodynamics and correlation functions of the
energy-momentum tensor in pure Yang-Mills gauge theory using the
energy-momentum tensor defined by the gradient flow and small flow time
expansion. Our results on thermodynamic observables are consistent with those
obtained by the conventional integral method. The analysis of the correlation
function of total energy supports the energy conservation. It is also addressed
that these analyses with gradient flow require less statistics compared with
the previous methods. All these results suggest that the energy-momentum tensor
can be successfully defined and observed on the lattice with moderate numerical
costs with the gradient flow.
