Using the Pollak-Grabert-Hanggi (PGH) weak-coupling approximation, the activated barrier crossing (ABC) problem is studied with a general memory friction kernel and for a general power-law potential added to a parabolic barrier. We focus on the recently discovered critical behavior of the rate for large memory friction correlation times. All the relevant critical exponents in different regimes of the strength of the friction are determined, and explicit expressions for the scaling function are obtained. We verify that the universality of exponents and amplitudes is applicable for this model within the PGH approximation. The results are compared with the results for canonical variational transition state theory (CVTST) recently obtained by us.