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Manfred comments on Logical thermodynamics: towards a theory of self-trusting uncertain reasoning - Less Wrong
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A general problem with big integrals in your definition of the abstract function P: Suppose that C is a statement whose shortest proof takes longer to write out than your agent has time. When asked for P(C), your abstract function P should not assign a number that your agent is physically incapable of assigning.
More concrete problem: since your energies are all greater than 0, proofs (evidence) with long length get weighted greater (are lower energy) than short proofs.
Evidence with long length is weighted less since it contributes less to the energy.
When computing probabilities at finite temperature to finite precision, it should be possible to ignore long evidence (at least evidence which is long wrt the given sentence). If the shortest evidence for a sentence is very long, it means its probability is "marginal" in some sense (since the total energy depends weakly on the value of the probability field at this statement). For sentences which cannot be decomposed into simpler sentences using propositional calculus operations, "marginal" probably means close to 1/2.
Oh, whoops. Yeah, my bad.