We prove that the multiplicity of a fixed eigenvalue $α$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to $μ_α n$ and $σ^2_α n$ respectively. It is also shown that $μ_α$ and $σ^2_α$ are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue $0$, the constants $μ_0$ and $σ^2_0$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.