Full-chip thermal monitoring is an important and challenging issue in today's microprocessor design. In this paper, we propose a new information-theoretic framework to quantitatively model the uncertainty of on-chip temperature variation by differential entropy. Based on this framework, an efficient optimization scheme is developed to find the optimal spatial locations for temperature sensors such that the full-chip thermal map can be accurately captured with a minimum number of on-chip sensors. In addition, several efficient numerical algorithms are proposed to minimize the computational cost of the proposed entropy calculation and optimization. As will be demonstrated by our experimental examples, the proposed entropy-based method achieves superior accuracy (1.4x error reduction) for full-chip thermal monitoring over prior art.