  The clockwork axion refers to a family of aligned multi-axion models that
lead to an exponential hierarchy between the scale of Peccei-Quinn symmetry
breaking and the scale of the axion decay constant. The clockworking can bring
the Peccei-Quinn-scale particles to within reach of collider experiments.In
this work we are interested in whether cosmological observations impose any new
constraints on the clockwork axion. If the universe reheats above the scale of
Peccei-Quinn breaking, then the ensuing cosmological phase transition produces
a network of topological defects, which have a qualitatively different behavior
from the string-wall network in the usual axion models. We estimate the relic
abundances of axion dark matter and dark radiation that arise from the emission
of axions by the defect network, and we infer a constraint on the scale of
Peccei-Quinn breaking and the mass spectrum. We find that the defect
contribution to the axion dark matter relic abundance is generally negligible.
However, the defect production of relativistic axion dark radiation becomes
significant if the scale of Peccei-Quinn symmetry breaking is larger than $100
\, {\rm TeV}$, and measurements of $\Delta N_{\rm eff}$ provide a new probe of
this class of models.
