We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the usual Cox ring. Then, we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the equivariant Cox ring is finitely generated in this case. Under a mild additional condition, we give a presentation by generators and relations of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The ordinary Cox ring also has a canonical structure of U-algebra, and we prove that the subalgebra of U-invariants is a finitely generated Cox ring of a variety of complexity one under the action of a torus. Using an earlier work from Hausen and Herppich, we obtain that this latter algebra is a complete intersection. Iteration of Cox rings has been introduced by Arzhantsev, Braun, Hausen and Wrobel in [1]. For log terminal quasicones with a torus action of complexity one, they proved that the iteration sequence is finite with a finitely generated factorial master Cox ring. We prove that the iteration sequence is finite for equivariant and ordinary Cox rings of normal rational G-varieties of complexity one satisfying a mild additional condition (e.g. complete varieties or almost homogeneous varieties with only constant invertible functions). In the almost homogeneous case, we prove that the equivariant and ordinary master Cox rings are finitely generated and factorial.