  In this paper, we consider eigenvalues to the following double phase problem
with unbalanced growth and indefinite weight, $$ -\Delta_p^a u-\Delta_q u
=\lambda m(x) |u|^{q-2}u \quad \mbox{in} \,\, \R^N, $$ where {$N \geq 2$},
{$1<p, q<N$, $p \neq q$}, ${a \in C^{0, 1}(\R^N, [0, +\infty))}$, $a \not\equiv
0$ and $m: \R^N \to \R$ is {an indefinite sign weight which may admit
nontrivial positive and negative parts}. Here $\Delta_q$ is the $q$-Laplacian
operator and $\Delta_p^a$ is the weighted $p$-Laplace operator defined by
$\Delta_p^a u:=\textnormal{div}(a(x) |\nabla u|^{p-2} \nabla u)$. The problem
can be degenerate, in the sense that the infimum of $a$ in $\R^N$ may be zero.
Our main results distinguish between the cases $p<q$ and $q<p$. In the first
case, we establish the existence of a {\it continuous} family of eigenvalues,
starting from the principal frequency of a suitable single phase eigenvalue
problem. In the latter case, we prove the existence of a {\it discrete} family
of positive eigenvalues, which diverges to infinity.
