  A (left) quandle is connected if its left multiplication group acts
transitively. In 2014, Eisermann introduced the concept of quandle coverings,
corresponding to so-called constant quandle cocycles that form a subset of
quandle cocycles. A connected quandle is said to be \emph{simply connected} if
it has no nontrivial coverings, or, equivalently, if its second constant
cohomology groups are trivial.
  In this paper we develop a combinatorial approach to constant cohomology.
Upon applying our theory, we prove that connected quandles that are affine over
cyclic groups are simply connected (extending a result of Gra\~{n}a for
quandles of prime size) and that finite doubly transitive quandles of order
different from $4$ are simply connected.
