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This paper outlines a thermodynamic theory of biological evolution. Beginning with a brief summary of the parallel histories of the modern evolutionary synthesis and thermodynamics, we use four physical laws and processes (the first and second laws of thermodynamics, diffusion and the maximum entropy production principle) to frame the theory. Given that open systems such as ecosystems will move towards maximizing dispersal of energy, we expect biological diversity to increase towards a level, Dmax, representing maximum entropic production (Smax). Based on this theory, we develop a mathematical model to predict diversity over the last 500 million years. This model combines diversification, post-extinction recovery and likelihood of discovery of the fossil record. We compare the output of this model with that of the observed fossil record. The model predicts that life diffuses into available energetic space (ecospace) towards a dynamic equilibrium, driven by increasing entropy within the genetic material. This dynamic equilibrium is punctured by extinction events, which are followed by restoration of Dmax through diffusion into available ecospace. Finally we compare and contrast our thermodynamic theory with the MES in relation to a number of important characteristics of evolution (progress, evolutionary tempo, form versus function, biosphere architecture, competition and fitness).