We have made long-term, high radio frequency observations of southern and equatorial active galactic nuclei (AGNs). After complementing these data with data from the literature, we have constructed the radio spectra of these sources and searched for sources with inverted spectra in the gigahertz range. We have identified 12 new sources with spectral shapes resembling those of the gigahertz-peaked spectrum (GPS) sources and eight other sources with inverted spectral parts in the gigahertz region. Several of these new GPS-source candidates have high (≥10 GHz) peak frequencies in the observer's frame, and they all exhibit strong long-term variability. We have also studied the variability behavior of known GPS sources, mainly quasar-type sources. All of the 14 sources included in our sample show moderate to extremely high radio variability, and at least 12 of them are also variable in the millimeter domain. Long-term monitoring of these sources shows that some of them have spectral shapes resembling those of classical variable flat-spectrum sources, indicating that some of these sources have been misidentified as GPS sources when only sparsely sampled data have been available. On the other hand, four of the variable sources show persistent GPS-type spectra at all stages of activity, suggesting that at least some of the GPS sources can be variable.