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This work proves an upper bound for the free energy of theSherrington-Kirkpatrick model and its generalizations in terms of theThouless-Anderson-Palmer (TAP) energy. The result applies to models withspherical or Ising spins and any mixed $p$-spin Hamiltonian with external fieldor with a non-linear spike term. The bound is expected to be tight to leadingorder at high temperature, and is non-trivial in the presence of an externalfield. For the proof a geometric microcanonical method is employed, in whichone covers the spin space with sets, each of which is centered at amagnetization vector $m$ and whose contribution to the partition function isbounded in terms of the TAP energy at $m$.

High temperature TAP upper bound for the free energy of mean field spin glasses