The well-known laxative effect of coffee can be attributed to its high acidity content, but is more likely due to the way coffee stimulates the stomach's production of gastrin, the main hormone that acts on the parietal cells inside the stomach to initiate acid production. It's this acidity increase via the release of gastrin that can lead to stomachache, heartburn, or indigestion. Furthermore, coffee has been shown to increase the production of cholecystokinin, another hormone that releases digestive enzymes and bile, which help regulate the process of pooping. However, it is worthwhile to note that there is no substantial research toward the exact compound in coffee that causes this reaction, nor is there any research regarding if this effect is innate to the coffee beans themselves or the method unto which they are turned into coffee (re: hot vs. cold extraction).
It's suspected that hot extraction of coffee pulls out different compounds than cold extraction, which can lead to differing effects on digestion. To that end, there is, however, research published on the differing acidity and caffeine levels of cold brew coffee versus regular coffee. According to research published in 2019* and 2020**, cold brew coffee has less acidity compared to regular coffee, as well as lower caffeine levels and antioxidants. (The same research also shows that the acidity of all coffee beverages decreased as degree of roast increased: the darkness of your roast may impact how your cold brew affects your digestion!)
Finally, there is plenty of anecdotal evidence out there that suggests cold brew coffee actually has a minimal impact on digestion compared to regular coffee, although of course everyone reacts differently. I personally experience less indigestion with cold brew than I do with regular coffee.
TL;DR (I know I wrote a lot; it's a subject I'm passionate about):
Cold brew coffee has decreased acidity and caffeine compared to regular coffee, although amounts also vary by degree of roast. As such, evidence suggests that cold brew coffee may actually have a lesser impact on the digestive system than regular coffee, although there is no significant research on the topic of gastrin release in cold brew vs. regular coffee, which is most likely the deciding factor for the majority of people's indigestion.
*"Effect of grinding, extraction time and type of coffee on the physicochemical and flavour characteristics of cold brew coffee," by Nancy Cordoba, et al. in Scientific Reports, Vol. 9 Article 8440 (2019). Very thorough examination into the difference between hot and cold extraction, with data regarding flavor profiles as well.
**"Physiochemical Characteristics of Hot and Cold Brew Coffee Chemistry: The Effects of Roast Level and Brewing Temperature on Compound Extraction," by Niny Z. Rao, et al. in Foods, Vol. 9 Issue 7 (2020). Straightforward comparison between cold and hot extraction with legible, quantified results.