  We study the scaling properties of the static quark potential and the ratio
of the critical temperature $T_c$ to the square root of the string tension
$\sigma$ in the SU(3) pure gauge theory using a renormalization group improved
action. We first determine the critical coupling $\beta_c$ on lattices with
temporal extension $N_t=3$, 4, and 6, and then calculate the static quark
potential at the critical couplings on lattices at zero temperature. We note
that the static quark potentials obtained are rotationally invariant with
errors of at most 1 - 2 % in all the three cases, and that the potential $V(R)$
in physical units scales in the whole region of $R$ investigated. The values of
$T_{c}/\sqrt{\sigma}$ for the three cases in the infinite volume limit are
identical within errors. We estimate the value in the continuum limit to be
$T_{c}/\sqrt{\sigma} = 0.656(4)$, which is slightly larger than the value in
the continuum limit from the one-plaquette action, 0.629(3).
