Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We define a new category in the following way: An object is an object $M$ of $D^b(X)$, together with an isomorphism $$\pi^*M\leftrightarrow \rho^*M$$ that satisfies the cocycle condition, where $\pi,\rho\colon G\times X\to X$ are the projection and the action maps. A morphism is a morphism of objects in $D^b(X)$ which is compatible with the action isomorphisms.

Note that this category is a special case of the category of equivariant objects: http://ncatlab.org/nlab/show/equivariant+object

Is this category equivalent to the equivariant derived category? If not, why is the equivariant derived category a better construction in this case?

PS: Bernstein and Lunts give among others the following definition of the equivariant derived category in their book "Equivariant sheaves and functors" (p.32):

Denote by $[X/G]$ the "action simplex" $\Delta(n)=G^{n-1}\times X$.

By a simplicial sheaf we mean a collection of sheaves $F^n$ on $\Delta(n)$ together with maps $\alpha_h: h^*F^m \rightarrow F^n$ for each map $h$ in $[X/G]$, which satisfy the cocycle condition: $$\alpha_{h' h}=\alpha_h \circ h^* \alpha_{h'}$$ Denote by $Sh([X/G])$ the category of simplicial sheaves and by $Sh_{eq}([X/G])$ the full subcategory where all $\alpha_h$ are isomorphisms.

The derived equivariant category $D^b_G(X)$ is then the full subcategory of $D^b(Sh([X/G]))$ consisting of objects, which have cohomology in $Sh_{eq}([X/G])$.