Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Let G be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the T-G-ideal of G-graded identities of a G-graded algebra in characteristic 0 and the generators of the T-GxZ2 -ideal of G x Z(2)-graded identities of its tensor product by the infinite-dimensional Grassmann algebra E endowed with the canonical grading have pairly the same degree. In this paper, we deal with Z(2) x Z(2)-graded identities of E-k* circle times E over an infinite field of characteristic p > 2, where E-k* is E endowed with a specific Z(2)-grading. We find identities of degree p + 1 and p + 2 while the maximal degree of a generator of the Z(2)-graded identities of E-k* is p if p > k. Moreover, we find a basis of the Z(2) x Z(2)-graded identities of E-k (*) circle times E and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the Z(2) x Z(2)-graded Gelfand-Kirillov (GK) dimension of E-k* circle times E.26611251140FAPESP [2013/06752-4, 2015/08961-5]CNPq - Brasil [305339/2013-3]"Para mulheres na Cieencia" (L'OREAL-ABC-UNESCO)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)