  In this paper we study quasilinear elliptic equations driven by the double
phase operator involving a Choquard term of the form
  \begin{align*}
  -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left(
\int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^\mu}\,\mathrm{d} y\right)f(x,u)
  \quad\text{in } \mathbb{R}^N,
  \end{align*}
  where $\mathcal{L}_{p,q}^{a}$ is the double phase operator given by
  \begin{align*}
  \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u +
a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in
W^{1,\mathcal{H}}(\mathbb{R}^N),
  \end{align*}
  $0<\mu<N$, $1<p<N$, $p<q<p+ \frac{\alpha p}{N}$, $0 \leq a(\cdot)\in
C^{0,\alpha}(\mathbb{R}^N)$ with $\alpha \in (0,1]$ and
$f\colon\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a continuous function
that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev
inequality, the Nehari manifold and variational tools, we prove the existence
of ground state solutions of such problems under different assumptions on the
data.
