We study the problem of continuous maintenance of range sum heat maps over dynamically updating data objects. The range sum (RS) here refers to the sum of the weights of the data objects enclosed by a given range (rectangle) R. Range sum problems are useful in spatio-Temporal data analytics and decision making processes. Recent studies on range sum problems focus on computing the MaxRS query, which finds a location to place a rectangle R such that its RS is maximized. In real applications, knowing only the location with the maximum RS may be insufficient, because decision making is a multi-factor process where maximizing the RS may just be one of the factors. It is also important to gain an overview of the RS distribution at different locations, so that decisions can be made based on global knowledge. We therefore propose to compute a range-sum heat map that visualizes the RS value for every location in a data space. Considering that data objects may be inserted into or removed from the data space dynamically, we further study the continuous maintenance of range-sum heat maps over dynamically updating data objects. We adapt algorithms to compute range-sum heat maps and to perform heat map updates. We build a demo system to showcase the usefulness of range sum heat maps and the effectiveness of the adapted algorithms.