Suppose \((C_t)_{t\geqslant 0}\) is the composition semigroup induced by a one-parameter semigroup \((\varphi _t)_{t\geqslant 0}\) of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of \((C_t)_{t\geqslant 0}\) acting on the weighted Bergman space induced by doubling weights, provided \((\varphi _t)_{t\geqslant 0}\) is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of \((C_t)_{t\geqslant 0}\) also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of \((C_t)_{t\geqslant 0}\) .
