  We show that regular semisimple Hessenberg varieties can have moduli. To be
precise, suppose $X$ is a regular semisimple Hessenberg variety of codimension
1 in the flag variety $G/B$, where $G$ is a simple algebraic group of rank $r$
over $\mathbb{C}$ and $B$ is a Borel subgroup. We show that the space
$\mathrm{H}^1(X,TX)$ of first order deformations of $X$ has dimension $r-1$
except in type $A_2$. (In type $A_2$, the Hessenberg varieties in question are
all isomorphic to the permutohedral toric surface, and
$\mathrm{dim}\mathrm{H}^1(X,TX)=0$.) Moreover, we show that the Kodaira-Spencer
map $\mathfrak{g}\to\mathrm{H}^1(X,TX)$ is onto, that the connected component
of the automorphism group of $X$ is the maximal torus of $G$, and that
$\mathrm{H}^i(X,TX)=0$ for $i\geq2$. Along the way, we prove several theorems
of independent interest about the cohomology of homogeneous vector bundles on
$G/B$.
  In type $A$, we can give an even more precise statement determining when two
codimension $1$ regular semisimple Hessenberg varieties in $G/B$ are
isomorphic. We also compute the automorphism groups explicitly in type
$A_{n-1}$ in the terms of stabilizer subgroups of the action of the symmetric
group $S_n$ on the moduli space $M_{0,n+1}$ of smooth genus $0$ curves with
$n+1$ marked points. Using this, we describe the moduli stack of the regular
semisimple Hessenberg varieties $X$ explicitly as a quotient stack of
$M_{0,n+1}$.
  We prove several analogous results for Hessenberg varieties in generalized
flag varieties $G/P$, where $P$ is a parabolic subgroup of $G$. In type $A$,
these results are used in the proofs of the results for $G/B$, but they are
also independently interesting because the associated moduli stacks are related
directly to the action of $S_n$ on $M_{0,n}$.
