While Nadarajah (2005) may have used the term 'generalised Normal' to describe a density that nests this form, there are more suitable names that extend far further back in time, and which accordingly seem much more appropriate.
In particular, I believe this should properly referred to as a Subbotin distribution (Subbotin 1923). Other later references include:
Diananda (1949)
Turner (1960)
Zeckhauser and Thompson (1970)
McDonald and Newey (1988)
Mineo and Ruggieri (2005)
The functional form given by Subbotin (1923) defines the pdf as:
$$f(x) = \frac{\alpha }{2 b \Gamma \left(\frac{1}{\alpha }\right)}\text{exp}\left[-\left|\frac{x}{b}\right|^{\alpha }\right]$$
Subbotin used parameter $b = 1/h$, but the functional form is otherwise identical to that given here. Here is a plot of the pdf with $b=2$, as parameter $\alpha$ varies:
http://www.tri.org.au/se/Subbotinpdfplot.png
In this form: $$Var(X) = \frac{b^2 \Gamma \left(\frac{3}{\alpha}\right)}{\Gamma \left(\frac{1}{\alpha}\right)}$$
Other names include: Box-Tiao distribution (McDonald and Newey 1988), and Power-Exponential (McDonald and Newey 1988, Johnson et al. 1995). Finally, it is worth noting that some economic papers inappropriately ascribe the name 'Subbotin distribution' to an Exponential-Power distribution that has a different functional form.