  Let $r\geq 2$ and $(X_i,G)$ $(i=1,\cdots,r)$ be topological dynamical systems
with $G$ being an infinite discrete amenable group. Suppose that
$\pi_i:(X_i,G)\to (X_{i+1},G)$ are factor maps and $0\leq w_i\leq 1$. In this
article, for $f\in C(X_1)$, we introduce the weighted topological pressure
$P^{\textbf{a}}(f,G)$ for higher dimensions (not only for $r=2$) of amenable
group actions. By using measure-theoretical theory, we establish a variational
principle as \begin{align*}
  P^{\textbf{a}}(f,G)=\sup_{\mu\in
\mathcal{M}^G(X_1)}\Big(\sum_{i=1}^rw_ih_{\mu_i}(X_i,G)+w_1\int_{X_1}fd\mu\Big),
\end{align*} where $\mu_i=\pi_{i-1}\circ\cdots\circ\pi_{1}\mu$ is the induced
$G$-invariant measure on $X_{i}$.
