Here is the sentence I always repeat to myself:

When replacement doesn't work look at the FullForm.

In[1]:= R = (a + b)/(c0 - c1 + g)^2;
FullForm[R]

Out[2]//FullForm= Times[Plus[a, b], Power[Plus[c0, Times[-1, c1], g], -2]]

For your example I want to replace that Power term with another term. But I want that new term to remain in the denominator so I still want that to have an exponent. So I try this

In[3]:= R = R /. Power[Plus[c0, Times[-1, c1], g], -2] -> A^-1

Out[3]= (a + b)/A

And that worked on the first try! That is often not the case.