  We show that the simplicity constraints, which define the dynamics of spin
foam models, imply, and are implied by, the first law of thermodynamics, when
the latter is applied to causal diamonds in the quantum spacetime. This result
reveals an intimate connection between the holographic nature of gravity, as
reflected by the Bekenstein entropy, and the fact that general relativity and
other gravitational theories can be understood as constrained topological field
theories.
  To state and derive this correspondence we describe causal diamonds in the
causal structure of spin foam histories and generalize arguments given for the
near horizon region of black holes by Frodden, Gosh and Perez and Bianchi. This
allows us to apply a recent argument of Jacobson to show that if a spin foam
history has a semiclassical limit described in terms of a smooth metric
geometry, that geometry satisfies the Einstein equations.
  These results suggest also a proposal for a quantum equivalence principle.
