Robot-assisted surgery (RAS), a type of minimally invasive surgery, is used in a variety of clinical surgeries because it has a faster recovery rate and causes less pain. Automatic video analysis of RAS is an active research area, where precise surgical tool detection in real time is an important step. However, most deep learning methods currently employed for surgical tool detection are based on anchor boxes, which results in low detection speeds. In this paper, we propose an anchor-free convolutional neural network (CNN) architecture, a novel frame-by-frame method using a compact stacked hourglass network, which models the surgical tool as a single point: the center point of its bounding box. Our detector eliminates the need to design a set of anchor boxes, and is end-to-end differentiable, simpler, more accurate, and more efficient than anchor-box-based detectors. We believe our method is the first to incorporate the anchor-free idea for surgical tool detection in RAS videos. Experimental results show that our method achieves 98.5% mAP and 100% mAP at 37.0 fps on the ATLAS Dione and Endovis Challenge datasets, respectively, and truly realizes real-time surgical tool detection in RAS videos.