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We compute the equivariant index of the twisted horizontal Dolbeault operatoron compact toric contact manifolds of Reeb type. The operator is elliptictransverse to the Reeb foliation and its equivariant index defines adistribution on the torus. Using the good cone condition, we show that thesymbol localises to the closed Reeb orbits corresponding to the edges of themoment cone and obtain an Atiyah-Bott-Lefschetz type formula for the index. Forthe horizontal Dolbeault operator, we obtain an expression for the index as asum over the lattice points of the moment cone, by applying an adaptation ofthe Lawrence-Varchenko polytope decomposition to rational polyhedral cones.

Equivariant Index on Toric Contact Manifolds