Lattice power of an algebra is a kind of its extension by a suitable complemented lattice. If the lattice is Boolean, this extension coincides with the Boolean power. In this paper, lattice powers of unary algebras are investigated. It turns out that any variety of unary algebras is preserved under the construction of lattice powers. It is proved that the lattice power of an algebra in such variety is isomorphic with the union of particular Boolean powers of the same algebra.