  In this paper we develop the homological properties of the $(\mathcal{L},
\mathcal{A})$-Gorenstein flat $R$-modules $\mathcal{GF}_{(\mathcal{F}(R),
\mathcal{A})}$ proposed by Gillespie. Where the class $\mathcal{A} \subseteq
\mathrm{Mod} (R^{op})$ sometimes corresponds to a duality pair $(\mathcal{L},
\mathcal{A})$. We study the weak global and finitistic dimensions that comes
with $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ and show that over a
$(\mathcal{L}, \mathcal{A})$-Gorenstein ring, the functor $-\otimes _R -$ is
left balanced over $\mathrm{Mod} (R^{op}) \times \mathrm{Mod} (R)$ by the
classes $\mathcal{GF}_{(\mathcal{F}(R^{op}), \mathcal{A})} \times
\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$. When the duality pair is
$(\mathcal{F} (R), \mathcal{FP}Inj (R^{op}))$ we recover the G. Yang's result
over a Ding-Chen ring, and we see that is new for $(\mathrm{Lev} (R),
\mathrm{AC} (R^{op}))$ among others.
