We study the stabilizability of a class of abstract parabolic equations of the form u′(t) + Au(t) + p(t)Bu(t) = 0, t ≥ 0 where the control p(·) is a scalar function, A is a self-adjoint operator on a Hilbert space X that satisfies A ≥ −σI, with σ > 0, and B is a bounded linear operator on X. Denoting by {λk}k∈N∗ and {φk}j∈N∗ the eigenvalues and the eigenfunctions of A, we show that the above system is locally stabilizable to the eigensolutions ψ j = e−λ j t φ j with doubly exponential rate of convergence, provided that the associated linearized system is null controllable. Moreover, we give sufficient conditions for the pair {A, B} to satisfy such a property, namely a gap condition for A and a rank condition for B in the direction φ j . We give several applications of our result to different kinds of parabolic equations.