In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even in the singular case) in terms of combinatorial data coming from the fan of the toric variety.
Abdó Roig-Maranges. "Morphic cohomology of toric varieties." Homology Homotopy Appl. 14 (1) 113 - 132, 2012.