  In this paper we compute the Casimir energy and force for massless scalar
fields endowed with hybrid boundary conditions, in the setting of the bounded
generalized cone. By using spectral zeta function regularization methods, we
obtain explicit expressions for the Casimir energy and force in arbitrary
dimensions in terms of the zeta function defined on the piston. Our general
formulas are, subsequently, specialized to the case in which the piston is
modelled by a $d$-dimensional sphere. In this particular situation, explicit
results are given for $d=2,3,4,5$.
