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>



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> Is this category equivalent to the equivariant derived category? If not, why is the equivariant derived category a better construction in this case?
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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 $Sh\_{eq}([X/G])$.



Is this a definition of equivariant derived category?
The relation is easier to understand if $G$ is a discrete group. Then your
definition is equivalent to an action of $G$ on an object $M$ in $D^b(X)$. This
means that for each $g\in G$ we have a morphism $\phi\_g\colon M\rightarrow M$ in
$D^b(X)$ with, and this is the important part, $\phi\_{gh}=\phi\_g\phi\_h$
*in* $D^b(X)$. Seen from the point of view of complexes (i.e., before we
pass to $D^b(X)$) this means that we have a homotopy (or something slightly
worse depending on what kind of complex $M$ is) $\phi\_{gh}\sim\phi\_g\phi\_h$.
Experience tells us that such actions up to homotopy is too weak a notion to be
useful (things are a little bit tricky as there are non-trivial situations when
such an action can in fact be replaced up to homotopy by a true action) and in
general one should demand an actual action of $G$ on the complex $M$. The same
goes (and in fact even more so) for morphisms, there may be too many morphisms
if you just look at morphisms in $D^b(X)$ that commute *in* $D^b(X)$ with
the action of $G$.


**Addendum**: Donu makes an excellent point and I just want to elaborate. Take an additive $G$-action on some $(\mathbb Z/p)^n$ which does not lift to an additive action on $(\mathbb Z/p^2)^n$. Then we may look at the morphism $(\mathbb Z/p)^n \to (\mathbb Z/p)^n[1]$ whose mapping cone is $(\mathbb Z/p^2)^n$. It will be a $G$-map $D^b(pt)$ but there is no compatible action of $G$ on a mapping cone.


