Abstract: Let R be an associative ring with an identity, and (L,A) be a complete duality pair. Firstly, we introduce the class GP⁽²⁾_L of Gorenstein homological modules with respect to the complete duality pair (L,A). Secondly,  we study some properties of GP⁽²⁾_L. Finally, we prove that GP⁽²⁾_L coincides with the class of Gorenstein (L,A)-projective modules with the help of some special classes of modules.

Key words: Gorenstein , (L,A)-projective module, stability, Gorenstein flat module, exact sequence, duality pair