The resolution of the P vs. NP problem, a cornerstone in computational theory, remains elusive despite extensive exploration through mathematical logic and algorithmic theory. This paper takes a novel approach by integrating information theory, thermodynamics, and computational complexity, offering a comprehensive landscape of interdisciplinary study. We focus on entropy, a concept traditionally linked with uncertainty and disorder, and reinterpret it to assess the complexity of computational problems. Our research presents a structured framework for establishing entropy profiles within computational tasks, enabling a clear distinction between P and NP-classified problems. This framework quantifies the 'information cost' associated with these problem categories, highlighting their intrinsic computational complexity. We introduce Entropy-Driven Annealing (EDA) as a new method to decipher the energy landscapes of computational problems, focusing on the unique characteristics of NP problems. This method proposes a differential thermodynamic profile for NP problems in contrast to P problems and explores potential thermodynamic routes for finding polynomial-time solutions to NP challenges. Our introduction of EDA and its application to complex computational problems like the Boolean satisfiability problem (SAT) and protein-DNA complexes suggests a potential pathway toward unraveling the intricacies of the P vs. NP problem.