Context. As the Square Kilometer Array (SKA) is expected to be operational in the next decade, investigations of the radio sky in the range of 100 MHz–10 GHz have become important for simulating SKA observations. In determining physical properties of galaxies from radio data, the radio spectral energy distribution (SED) is often assumed to be described by a simple power law, usually with a spectral index of 0.7 for all sources. Even though radio SEDs have been shown to exhibit deviations from this assumption, both in differing spectral indices and complex spectral shapes, it is often presumed that their individual differences can be canceled out in large samples. Aims. Since the average spectral index around 1 GHz (observed-frame) is important for determining physical properties of large samples of galaxies, we aim to test whether individual differences in the spectra of radio-identified active galactic nuclei align with the simple assumption of α  = 0.7 and test the evolution of the parameters of the synchrotron aging model with redshift and radio luminosity. Methods. We use a sample of 744 radio-excess active galactic nuclei (RxAGN), defined as those that exhibit more than a 3 σ radio luminosity excess with respect to the value expected only from the contribution from star formation, out to z  ∼ 4. We constructed their average radio SED by combining Very Large Array (VLA) observations of the COSMOS field at 1.4 GHz and 3 GHz with Giant Meterwave Radio Telescope (GMRT) observations at 325 MHz and 610 MHz. To account for nondetections in the GMRT maps, we employed the survival analysis technique. We binned the RxAGN sample into luminosity- and redshift-complete subsamples. In each bin, we constrained the shape of the average radio SED by fitting a broken power-law model. Results. We find that the RxAGN sample can be described by a spectral index of α 1  = 0.28 ± 0.03 below the break frequency ν b  = (4.1 ± 0.2) GHz and α 2  = 1.16 ± 0.04 above it, while a simple power-law model, capturing fewer spectral features, yields a single spectral index of 0.64 ± 0.07. By binning in 1.4 GHz of radio luminosity and redshift, we find that the power-law spectral index is positively correlated with redshift and that the broken power-law spectral index above 4 GHz is positively correlated with both the redshift and source size. By selecting sources with sizes less than 1 kpc, we find a subsample of flat-spectrum sources, which can be described by a spectral index of α  = 0.41 ± 0.07 and a broken power-law spectral index of α 1  = 0.1 ± 0.1 ( α 2  = 0.55 ± 0.09) below (above) a break frequency of ν b  = (2.7 ± 0.5) GHz. Conclusions. We have constrained the radio SED for a sample of RxAGN in the COSMOS field using available VLA and GMRT data, corresponding to the rest-frame frequency range from ∼0.3 GHz to ∼10 GHz. We describe our derived average radio SED of RxAGN using power-law and broken power-law models, yielding a radio SED that steepens above ∼4 GHz.