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We study the scaling properties of the static quark potential and the ratioof 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 improvedaction. We first determine the critical coupling $\beta_c$ on lattices withtemporal extension $N_t=3$, 4, and 6, and then calculate the static quarkpotential at the critical couplings on lattices at zero temperature. We notethat the static quark potentials obtained are rotationally invariant witherrors 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 areidentical 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 inthe continuum limit from the one-plaquette action, 0.629(3).Comment: 32 pages of Latex, 13 PostScript figures. Replaced version containing  extended finite size scaling analyses on $15^{3}\times3$ and $20^{3}\times4$  lattice

Scaling in SU(3) pure gauge theory with a renormalization-group-improved action