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For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential 
$-t\log J^{u}$, where 
$t\in \mathbb{R}$ is in a certain large interval and 
$J^{u}$ denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.
