Range counting query is the fundamental task for data analysis and data mining in Internet of Things (IoT). However, it poses a threat to the data privacy of data contributors, which is exacerbated by evolving data in particular. Several studies focus on range counting query on a timestamp over the finite evolving data, and thus are not applicable to the longitudinal range counting query on the infinite evolving data. To this end, we propose thePrivate andEffectiveRange counting query overEvolving data (PERE) that supports both the finite evolving data and the infinite evolving data, and is applicable for both the range counting query on a specific timestamp and the longitudinal range counting query. Specifically, we first design a Private Infinite Update Framework for IoT evolving data while providing meaningful privacy protection. The framework is coupled with a general and practical data evolution paradigm. Then, we propose a Optimized Frequency Perturbation consisting of an enhanced frequency oracle protocol and random sampling attribute. On this basis, we further propose a novel Adaptive Interval Merging mechanism that dynamically considers all potential interval consolidation possibilities and the reasonable selection of intervals for merging, to balance non-uniform error and noise error. Thereafter, we further reduce the estimated error in query results by Frequency Adjustment that consists of Norm-Sub and weighted average process. Last, we theoretically prove that the proposed PERE satisfies Local Differential Privacy (LDP), that the query results of PERE are unbiased, and that the variance of the query results is desirable. Furthermore, the extensive experiments on multiple real-world and synthetic datasets validate the effectiveness of PERE, as well as its advantages over the state-of-the-art works in answering range counting queries of evolving data.