  We present and study the concept of $m$-periodic Gorenstein objects relative
to a pair $(\mathcal{A,B})$ of classes of objects in an abelian category, as a
generalization of $m$-strongly Gorenstein projective modules over associative
rings. We prove several properties in some cases where $(\mathcal{A,B})$
satisfies certain homological conditions, like for instance when
$(\mathcal{A,B})$ is a GP-admissible pair. Connections to Gorenstein objects
and Gorenstein homological dimensions relative to these pairs are also
established.
