We consider a cognitive radio-based Internet-of-Things (CR-IoT) network consisting of one primary IoT (PIoT) system and one secondary IoT (SIoT) system. The IoT devices of both the PIoT and the SIoT respectively monitor one physical process and send randomly generated status updates to their associated access points (APs). The timeliness of the status updates is important as the systems are interested in the latest condition (e.g., temperature, speed and position) of the IoT device. In this context, two natural questions arise: (1) How to characterize the timeliness of the status updates in CR-IoT systems? (2) Which scheme, overlay or underlay, is better in terms of the timeliness of the status updates. To answer these two questions, we adopt a new performance metric, named the age of information (AoI). We analyze the average peak AoI of the PIoT and the SIoT for overlay and underlay schemes, respectively. Simple asymptotic expressions of the average peak AoI are also derived when the PIoT operates at high signal-to-noise ratio (SNR). Based on the asymptotic expressions, we characterize a critical generation rate of the PIoT system, which can determine the superiority of overlay and underlay schemes in terms of the average peak AoI of the SIoT. Numerical results validate the theoretical analysis and uncover that the overlay and underlay schemes can outperform each other in terms of the average peak AoI of the SIoT for different system setups.