Object goal navigation (ObjectNav) is a fundamental task in embodied AI, requiring an agent to locate a target object in previously unseen environments. This task is particularly challenging because it requires both perceptual and cognitive processes, including object recognition and decision-making. While substantial advancements in perception have been driven by the rapid development of visual foundation models, progress on the cognitive aspect remains constrained, primarily limited to either implicit learning through simulator rollouts or explicit reliance on predefined heuristic rules. Inspired by neuroscientific findings demonstrating that humans maintain and dynamically update fine-grained cognitive states during object search tasks in novel environments, we propose CogNav, a framework designed to mimic this cognitive process using large language models. Specifically, we model the cognitive process using a finite state machine comprising fine-grained cognitive states, ranging from exploration to identification. Transitions between states are determined by a large language model based on a dynamically constructed heterogeneous cognitive map, which contains spatial and semantic information about the scene being explored. Extensive evaluations on the HM3D, MP3D, and RoboTHOR benchmarks demonstrate that our cognitive process modeling significantly improves the success rate of ObjectNav at least by relative 14% over the state-of-the-arts.