  For arbitrary connected reductive group G we consider the motivic integral
over the arc space of an arbitrary Q-Gorenstein horospherical G-variety
associated with a colored fan and prove a formula for the stringy E-function of
a horospherical variety X which generalizes the one for toric varieties. We
remark that in contrast to toric varieties the stringy E-function of a
Gorenstein horospherical variety X may be not a polynomial if some cones in the
fan of X have nonempty sets of colors. Using the stringy E-function, we can
formulate and prove a new smoothness criterion for locally factorial
horospherical varieties. We expect that this smoothness criterion holds for
arbitrary spherical varieties.
