Motivated by the study of the Donaldson-Thomas theory on toric CY 3-folds I will reconsider application of Duistermaat-Heckman formula for non-compact toric Kahler manifolds. I will review the calculation of equivariant volumes for non-compact manifolds and explain some underlying issues. I will derive the set of finite-difference equations obeyed by equivariant volumes and their quantum versions. The talk is based on my joint paper with Nikita Nekrasov and Nicolo Piazzalunga (with appendix by Michele Vergne).