Define a one-parameter exponential family as a family of densities of the form $f_\theta(x)=\exp(\eta(\theta)T(x) + \xi(\theta))h(x)$ where $T(x)$ and $h(x)$ are Borel functions, $\theta\in\Theta\subset\mathbb R$ and $\eta$ and $\xi$ are real-valued functions defined on $\Theta$.
Double exponential distribution is a distribution having the density $p_\theta(x)= \frac{1}{2}\exp(-|x - \theta|)$ for $\theta\in\mathbb R$.
I am looking for a simple proof of the theorem in the title. I found a proof in the book of Shao "Mathematical Statistics. Exercises and Solutions." but it uses a more general definition of exponential families and doesn't show why the classes are not compatible. What is the special feature of $p_\theta(x)$ that makes the representation as exponential family impossible?