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The Fock-Bargmann-Hartogs domain $D_{n, m}(\mu)$ is defined by $$ D_{n,m}(\mu):=\{(z, w)\in\mathbb{C}^{n}\times\mathbb{C}^m:\Vert w\Vert^2<e^{-\mu\Vert z \Vert^2}\},$$ where $\mu>0.$ The Fock-Bargmann-Hartogsdomain $D_{n, m}(\mu)$ is an unbounded strongly pseudoconvex domain with smoothreal-analytic boundary. In this paper, we first compute the weighted Bergmankernel of $D_{n, m}(\mu)$ with respect to the weight $(-\rho)^{\alpha}$, where$\rho(z,w):=\|w\|^2-e^{-\mu \|z\|^2}$ is a defining function for $D_{n,m}(\mu)$ and $\alpha>-1$. Then, for $p\in [1,\infty),$ we show that thecorresponding weighted Bergman projection $P_{D_{n, m}(\mu), (-\rho)^{\alpha}}$is unbounded on $L^p(D_{n, m}(\mu), (-\rho)^{\alpha})$, except for the trivialcase $p=2$. In particular, this paper gives an example of an unbounded stronglypseudoconvex domain whose ordinary Bergman projection is $L^p$ irregular when$p\in [1,\infty)\setminus\{2\}$. This result turns out to be completelydifferent from the well-known positive $L^p$ regularity result on boundedstrongly pseudoconvex domain.

$L^{p}$ regularity of weighted Bergman projection on Fock-Bargmann-Hartogs domain