Persistence entropy (PE), one of the most widely used summary statistics in topological data analysis, exhibits a universal scaling PE(n) = alpha ln(n) + beta across a broad range of constructions. We show that the slope alpha equals delta, the exponent governing how the number of persistence bars m scales with input size: m ~ n^delta. This bar count law is verified across 14 constructions spanning random matrix eigenvalue spacings, Erdos-Renyi graphs, complete graph minimum spanning trees, random point clouds, manifold samples, Brownian motion paths, and Ginibre eigenvalue point clouds. We derive an exact closed-form PE for the Rips complex of uniform points on S^1 (0.04% accuracy at n=1000), show via Kac-Rice theory that the eigenvalue process PD has m ~ n^{3/2} bars from level repulsion (verified to 0.46% across 11 matrix sizes), and establish a central limit theorem for PE of geometric constructions. A previously reported anomalous PE slope (alpha = 1.455) is identified as a log-base artifact.