  We study the cohomology of families of $(\varphi,\Gamma)$-modules with
coefficients in pseudoaffinoid algebras. We prove that they have finite
cohomology, and we deduce an Euler characteristic formula and Tate local
duality. We classify rank-$1$ $(\varphi, \Gamma)$-modules and deduce that
triangulations of pseudorigid families of $(\varphi,\Gamma)$-modules can be
interpolated, extending a result of [KPX14]. We then apply this to study
extended eigenvarieties at the boundary of weight space, proving in particular
that the eigencurve is proper at the boundary and that Galois representations
attached to certain characteristic $p$ points are trianguline.
