  This paper concerns the study of regular Fourier hypergroups through
multipliers of their associated Fourier algebras. We establish hypergroup
analogues of well-known characterizations of group amenability, introduce a
notion of weak amenability for hypergroups, and show that every discrete
commutative hypergroup is weakly amenable with constant 1. Using similar
techniques, we provide a sufficient condition for amenability of hypergroup
Fourier algebras, which, as an immediate application, answers one direction of
a conjecture of Azimifard--Samei--Spronk [J. Funct. Anal. 256(5) 1544-1564,
2009] on the amenability of $ZL^1(G)$ for compact groups $G$. In the final
section we consider Fourier algebras of hypergroups arising from compact
quantum groups $\mathbb{G}$, and in particular, establish a completely
isometric isomorphism with the center of the quantum group algebra for compact
$\mathbb{G}$ of Kac type.
