In this paper, we consider the following double phase problem with combined power nonlinearities and multiple potentials \(\begin{aligned} & -\Delta _p u-\Delta _q u+V(x)(|u|^{p-2}u+|u|^{q-2}u)\\ & \quad =\lambda |u|^{p-2}u+\mu (x)|u|^{m-2}u+|u|^{l-2}u,\quad \text {in}\quad {\mathbb {R}}^N \end{aligned}\) having prescribed mass \(\begin{aligned} \int _{{\mathbb {R}}^N}|u|^p\,dx=c^p, \end{aligned}\) where \(1<p<q<N,\) \(p<m<\frac{Np}{N-p},\) \(\frac{q(N+p)}{N}<l<\frac{Np}{N-p},\) V(x) is an external potential vanishing at infinity, \(\mu (x)\ge 0\) is a competing potential, and the parameter \(\lambda \in {\mathbb {R}}\) appears as a Lagrange multiplier. Under some mild assumptions on V(x) and \(\mu (x),\) for the \(L^p\) -subcritical case \(m\in \left( p,\frac{p(N+p)}{N}\right) ,\) we show that there exists \(c_0>0\) such that the normalized solution with negative energy can be obtained when \(c\in (0,c_0)\) . This is achieved by analyzing the relationship between the energy of double phase problem and its limit problem. For the \(L^p\) -critical case \(m=\frac{q(N+p)}{N},\) by limiting the range of parameters, we find the existence of positive ground state normalized solutions with the aid of the Pohozaev constraint and using the properties of the energy in relationship with its limit problem. For the \(L^p\) -supercritical case \(m\in \left( \frac{q(N+p)}{N},\frac{Np}{N-p}\right) ,\) we get the existence of positive ground state normalized solutions for the double phase problem by estimating the energy of the double phase problem and its limit problem and using the Pohozaev constraint.
