In this work, we study the multiplicity of positive solutions for a critical fractional Choquard equation with perturbation on a bounded star-shaped domain. The problem is given by
\((-\Delta)^s u = \lambda u + \alpha |u|^{p-2}u + \left(\int_{\Omega} \frac{|u(y)|^{2^*_{\mu,s}}}{|x-y|^\mu} dy \right)|u|^{2^*_{\mu,s}-2}u\) in \(\Omega\), with \(u>0\) in \(\Omega\), \(u=0\) in \(\mathbb{R}^N \setminus \Omega\), and \(\int_{\Omega} |u|^2 dx = d\).

Using variational methods, we prove the existence of multiple solutions. In particular, a first solution is obtained via minimization on a suitable constraint set, while a second solution is derived using a uniform mountain pass theorem.