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The local quench of a Fermi gas, giving rise to the Fermi edge singularity and the Anderson orthogonality catastrophe, is an analytically tractable out-of-equilibrium problem in condensed matter. It describes the generic physics that occurs when a localized scattering potential is suddenly introduced in a Fermi sea, leading to a brutal disturbance of its quantum state. It has recently been proposed that the effect could be efficiently simulated in a controlled manner using the tunability of ultra-cold atoms. In this work, we analyze the quench problem in a gas of trapped ultra-cold fermions from a thermodynamic perspective using the full statistics of the so-called work distribution. The work statistics are shown to provide an accurate insight into the fundamental physics of the process.