  We study an eigenvalue problem in the framework of double phase variational
integrals and we introduce a sequence of nonlinear eigenvalues by a minimax
procedure. We establish a continuity result for the nonlinear eigenvalues with
respect to the variations of the phases. Furthermore, we investigate the growth
rate of this sequence and get a Weyl-type law consistent with the classical law
for the $p$-Laplacian operator when the two phases agree.
