  We prove continuity results for new stability thresholds related to uniform
K-stability and deduce that uniform K-stability is an open condition in the
K\"ahler cone of any compact K\"ahler manifold, thus establishing an
algebro-geometric counterpart to a classical result of LeBrun-Simanca for
constant scalar curvature (cscK) metrics. This settles a folklore conjecture in
the field, and in particular implies openness of uniform K-stability for smooth
polarized varieties. Moreover, it strengthens evidence supporting the uniform
version of the Yau-Tian-Donaldson conjecture for arbitrary polarizations,
including the case of irrational polarizations and non-projective K\"ahler
manifolds. As a key tool we introduce a new norm on test configurations and
establish estimates for non-archimedean energy functionals in terms of this
norm. This leads to new characterizations of uniform K-stability by restricting
to test configurations that satisfy certain uniform bounds. As a byproduct we
obtain continuity results for a stability threshold related to non-archimedean
entropy and deduce openness of uniform J-stability, as well as openness of
J-stability in the projective case.
