The derived category with respect to $\boldsymbol{\mathcal{G}(\mathcal{X})}$

Let $\mathcal{A}$ be an abelian category with enough projective objects and $(\mathcal{X},~\mathcal{Y})$ be a complete and hereditary cotorsion pair in $\mathcal{A}$. We introduce the Gorenstein $\mathcal{X}$-derived category, denoted by $\mathbf{D}_{\mathcal{G}(\mathcal{X})}{(\mathcal{A})}$, and investigate $\mathbf{D}_{\mathcal{G}(\mathcal{X})}{(\mathcal{A})}$ from different perspectives. When $(\mathcal{X},~\mathcal{Y})$ is served as a special cotorsion pair, we obtain some known derived categories, such as Gorenstein derived categories, Gorenstein flat derived categories and so on. The bounded Gorenstein $\mathcal{X}$-derived category $\mathbf{D}^{b}{_{\mathcal{G}(\mathcal{X})}{(\mathcal{A})}}$ and bounded derived category $\mathbf{D}^{b}{(\mathcal{A})}$ are described via the homology category $\mathbf{K}^{-,gxb}{(\mathcal{G}(\mathcal{X}))}$ related to $\mathcal{G}(\mathcal{X})$ and some triangle equivalences are given.