In this manuscript we consider the set of Dyck paths equipped with the uniform measure, and we study the statistical properties of a deformation of the observable "area below the Dyck path" as the size $N$ of the path goes to infinity. The deformation under analysis is apparently new: while usually the area is constructed as the sum of the heights of the steps of the Dyck path, here we regard it as the sum of the lengths of the connected horizontal slices under the path, and we deform it by applying to the lengths of the slices a positive regular function $\omega(\ell)$ such that $\omega(\ell) \sim \ell^p$ for large argument. This shift of paradigm is motivated by applications to the Euclidean Random Assignment Problem in Random Combinatorial Optimization, and to Tree Hook Formulas in Algebraic Combinatorics. For $p \in \mathbb{R}^+ \smallsetminus \left\{ \frac{1}{2}\right\}$, we characterize the statistical properties of the deformed area as a function of the deformation function $\omega(\ell)$ by computing its integer moments, finding a generalization of a well-known recursion for the moments of the area-Airy distribution, due to Tak\'acs. Most of the properties of the distribution of the deformed area are \emph{universal}, meaning that they depend on the deformation parameter $p$, but not on the microscopic details of the function $\omega(\ell)$. We call \emph{$p$-Airy distribution} this family of universal distributions.