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The article considers the optimization of eigenvalues in electromagnetic cavities by means of shape variations. The field distribution and its frequency in a radio‐frequency cavity are governed by Maxwell's eigenvalue problem. To this end, we utilize a mixed formulation by Kikuchi (1987) and a mixed finite element discretization by means of Nédélec and Lagrange elements. The shape optimization is based on the method of mappings, where a Piola transformation is utilized to assert conformity of the mapped spaces. We derive the derivatives by the use of adjoint calculus for the constraining Maxwell eigenvalue problem. In two numerical examples, we demonstrate the functionality of this method.