  We present a lattice calculation of the entropy density $s/T^{3}$ and speed
of sound $c_{s}^{2}$ of gluedynamics near the critical temperature, $T_{c}$, in
the deconfined phase. By exploring the temperature dependence of entropy
density in this region, we aim to analyse the significant discrepancies between
the previous computations. The calculation of entropy density is carried out by
numerical simulations of $O(a^{4})$ mean-field improved energy-momentum tensor
(EMT) of SU(3) gauge theory on the lattice. We expand on reaching $O(a^{4})$
improvement using tadpole-improved Symanzik action. The entropy density is
calculated directly from the expectation value of the space-time component of
the improved EMT in the presence of shifted boundary conditions at several
lattice spacings ($a \approx 0.043 - 0.012$ fm). The absence of ultraviolet
divergences and the minimal finite-size effects allow for the precision
determination of the entropy density and its extrapolation to the continuum
limit. As expected, the resulting entropy density displays the expected
behaviour of rapid increase near the critical temperature in the deconfined
phase followed by a slow increase in $2T_{c}\leq T\leq 3T_{c}$ region,
suggesting a logarithmic dependence on the temperature. A quantitative
comparison of $s/T^{3}$ shows good agreement with Pade approximation and
lattice results of previous high-precision data obtained using the gradient
flow method. We observe that at temperatures of about $3T_{c}$, deviations of
entropy density from the Stefan-Boltzmann value for a free theory are about
10$\%$. It is shown that the speed of sound in SU(3) gluedynamics is found to
be $c_{s}^{2}\leq 0.333$ in the temperature region $1.06T_{c}\leq T\leq
3.05T_{c}$ explored in this study. The results are found to agree with the
corresponding analytic and numerical estimates.
