Apart from the suspected violation of the CPT invariance, we might expect if the measurements of antihydrogen atoms provide testing Weak Equivalence Principle (WEP) in the gravitational phenomena. We start with how its violation can be related to the expected idea of unification of particle physics and gravitation, an attempt beyond the standard theories, including general relativity of Einstein. A particular emphasis will be placed on the issue of an accelerating universe, a rather recent development since nearly 10 years ago, suggesting a strong motivation toward attempts beyond the conventional concepts of the traditional cosmology. We face today's version of the cosmological constant problem. A candidate of the new ingredient appears to be provided by a scalar field, sometimes under the names like quintessence or dark-energy. In this article, we discuss the subject from a point of view of a more theoretical approach based on the scalar-tensor theory of gravitation. By exploiting the concepts of conformal transformations and conformal frames unique to this type of approach, we show that a successful understanding of the observed cosmological acceleration entails an unexpected outcome of breakdown of WEP which may show up as a distinct behavior between hydrogen and antihydrogen in the gravitational influence, from a further wider perspective including a vector field as well. We intended to introduce most of the new concepts as plainly and briefly as possible, according to the nature of the talk. On the other hand, however, we find it also useful to go into some more details, depending on the interests of individuals among the audience. For a compromise, we decided to add a number of footnotes which were not delivered during the presentation, but might be helpful in the proceedings published later.
