In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain \(\begin{aligned} \begin{array}{c} (-\Delta )^s u = \lambda u +\alpha |u|^{p-2}u+ \left( \int \limits _{\Omega } \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu }\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u\; \text {in} \; \Omega ,\\ u>0\; \text {in}\; \Omega ,\;\\ u = 0\; \text {in} \; {\mathbb {R}}^{N}\backslash \Omega , \\ \int _{\Omega }|u|^2dx=d, \end{array} \end{aligned}\) where \(s\in (0,1), N>2s\) , \(\alpha \in {\mathbb {R}}\) , \(d>0\) , \(2<p<2^*_s:=\frac{2N}{N-2s}\) , and \(2^{*}_{\mu ,s}:=\frac{2N-\mu }{N-2s}\) represents the fractional Hardy-Littlewood-Sobolev critical exponent. Using the minimization technique over an appropriate set and the uniform mountain pass theorem, we prove the existence of first and second solutions, respectively.
