  We measure correlation functions of the nonperturbatively renormalized
energy-momentum tensor in $N_f=2+1$ full QCD at finite temperature by applying
the gradient flow method both to the gauge and quark fields. Our main interest
is to study the conservation law of the energy-momentum tensor and to test
whether the linear response relation is properly realized for the entropy
density. By using the linear response relation we calculate the specific heat
from the correlation function. We adopt the nonperturbatively improved Wilson
fermion and Iwasaki gauge action at a fine lattice spacing $=0.07$ fm. In this
paper the temperature is limited to a single value $T=232$ MeV. The $u$, $d$
quark mass is rather heavy with $m_\pi/m_\rho=0.63$ while the $s$ quark mass is
set to approximately its physical value.
