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A (graph) property P is a class of simple finite graphs closed under isomorphisms. In this paper we consider generalizations of sum list colorings of graphs with respect to properties P. If to each vertex v of a graph G a list L(v) of colors is assigned, then in an (L, P)-coloring of G every vertex obtains a color from its list and the subgraphs of G induced by vertices of the same color are always in P. The P-sum choice number χ P sc (G) of G is the minimum of the sum of all list sizes such that, for any assignment L of lists of colors with the given sizes, there is always an (L, P)-coloring of G. We state some basic results on monotonicity, give upper bounds on the Psum choice number of arbitrary graphs for several properties, and determine the P-sum choice number of specific classes of graphs, namely, of all complete graphs, stars, paths, cycles, and all graphs of order at most 4.