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Aims.We introduce the TELAMON program which is using the Effelsberg 100-m telescope to monitor the radio spectra of active galactic nuclei (AGN) under scrutiny in astroparticle physics, specifically TeV blazars and candidate neutrino-associated AGN. Here, we present and characterize our main sample of TeV-detected blazars. Methods.We analyzed the data sample from the first ∼2.5 yr of observations between August 2020 and February 2023 in the range from 14 GHz to 45 GHz. During this pilot phase, we observed all 59 TeV-detected blazars in the Northern Hemisphere (i.e., Dec > 0°) known at the time of observation. We discuss the basic data reduction and calibration procedures used for all TELAMON data and introduce a sub-band averaging method used to calculate average light curves for the sources in our sample. Results.The TeV-selected sources in our sample exhibit a median flux density of 0.12 Jy at 20 mm, 0.20 Jy at 14 mm, and 0.60 Jy at 7 mm. The spectrum for most of the sources is consistent with a flat radio spectrum and we found a median spectral index (S(ν)∝ν^α) ofα = −0.11. Our results on flux density and spectral index are consistent with previous studies of TeV-selected blazars. Compared to the GeV-selected F-GAMMA sample, TELAMON sources are significantly fainter in the radio band. This is consistent with the double-humped spectrum of blazars being shifted towards higher frequencies for TeV-emitters (in particular for high-synchrotron peaked BL Lac type objects), which results in a lower radio flux density. The spectral index distribution of our TeV-selected blazar sample is not significantly different from the GeV-selected F-GAMMA sample. Moreover, we present a strategy to track the light curve evolution of sources in our sample for future variability and correlation analysis.