  We employ renormalization group (RG) summation techniques to obtain portions
of Laplace QCD sum rules for scalar gluon currents beyond the order to which
they have been explicitly calculated. The first two of these sum rules are
considered in some detail, and it is shown that they have significantly less
dependence on the renormalization scale parameter $\mu^2$ once the RG summation
is used to extend the perturbative results. Using the sum rules, we then
compute the bound on the scalar glueball mass and demonstrate that the 3 and
4-Loop perturbative results form lower and upper bounds to their RG summed
counterparts. We further demonstrate improved convergence of the RG summed
expressions with respect to perturbative results.
