  We study local and global higher integrability properties for quasiminimizers
of a class of double-phase integrals characterized by nonstandard growth
conditions. We work purely on a variational level in the setting of a metric
measure space with a doubling measure and a Poincar\'e inequality. The main
novelty is an intrinsic approach to double-phase Sobolev-Poincar\'e
inequalities.
