  We characterize ultrafilter convergence and ultrafilter compactness in
linearly ordered and generalized ordered topological spaces. In such spaces,
and for every ultrafilter $D$, the notions of $D$-compactness and of
$D$-pseudocompactness are equivalent. Any product of initially
$\lambda$-compact generalized ordered topological spaces is still initially
$\lambda$-compact. On the other hand, preservation under products of certain
compactness properties are independent from the usual axioms for set theory.
