In Beilinson, Ginzburg, and Soergel, "Koszul Duality Patterns in Representation Theory" (comment 3.4), the authors outline a spectral sequence as follows:

Given a filtered complex algebraic variety $X=X_0 \supset \cdots \supset X_r = \varnothing$ and $\mathcal{F} \in D(X)$, $\mathbb{H}^{\bullet} (\mathcal{F})$ is the limit of the spectral sequence with $E_1$-term $E_1 ^{p,q} = \mathbb{H}_{X_p - X_{p+1}} ^{p+q} (\mathcal{F})$.

I believe that the analogous result is true in the T-equivariant derived category. Does anyone know of a reference where I might find such a statement?