  While intersection cohomology is stable under small resolutions, both
ordinary and intersection cohomology are unstable under smooth deformation of
singularities. For complex projective algebraic hypersurfaces with an isolated
singularity, we show that the first author's cohomology of intersection spaces
is stable under smooth deformations in all degrees except possibly the middle,
and in the middle degree precisely when the monodromy action on the cohomology
of the Milnor fiber is trivial. In many situations, the isomorphism is shown to
be a ring homomorphism induced by a continuous map. This is used to show that
the rational cohomology of intersection spaces can be endowed with a mixed
Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary
cohomology of the singular hypersurface.
