We optimize the Age of Information (AoI) in random access networks using the age-threshold slotted ALOHA (TSA) protocol. The network comprises multiple source-destination pairs, where each source sends a sequence of status update packets to its destination over a shared spectrum. The TSA protocol stipulates that a source node must remain silent until its AoI reaches a predefined threshold, after which the node accesses the radio channel with a certain probability. We derive analytical expressions for the transmission success probability and time-average AoI using stochastic geometry tools. Subsequently, we obtain closed-form expressions for the optimal update rate and age threshold that minimize the time-average AoI. In addition, we establish a scaling law for the time-average AoI in random access networks, revealing that the optimal time-average AoI increases linearly with the deployment density. Notably, the growth rate under TSA is half of that under conventional slotted ALOHA.