  We derive variational formulas for the total Q-prime curvature under the
deformation of strictly pseudoconvex domains in a complex manifold. We also
show that the total Q-prime curvature agrees with the renormalized volume of
such domains with respect to the complete Einstein-K\"ahler metric. In the
appendix, by Rod Gover and the first author, we study the property of CR
invariant differential operator of order 2(n+3) on 2n+1 dimensional CR
manifolds that appears in the second variational formula of the total Q-prime
curvature.
