The binding energy, oscillator strengths, and absorption coefficients (ACs) for double ring-shaped quantum dots (DRSQDs) are theoretically investigated within the framework of effective mass approximation. The Schrödinger equation is solved by a variational method in order to compute the energy levels of a hydrogenic impurity in a gallium aresenide DRSQD. Obtained results have revealed the dependence of the binding energy on the dot radius and the potential parameters B and C. The energy values in the absence and presence of impurity are compared with the literature for spherical harmonic oscillator. The binding energy decreases as the quantum dot radius and the potential parameters increase. Also, it is found that the oscillator strengths and the linear and nonlinear optical ACs are quite sensitive to the potential parameters. The peaks of AC exhibit blueshift with B and C increasing.