In this paper, we introduce a weak form of amenability on topological semigroups that we call φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-amenability, where φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} is a character on a topological semigroup. Some basic properties of this new notion are obtained and by giving some examples, we show that this definition is weaker than the amenability of semigroups. As a noticeable result, for a topological semigroup S, it is shown that if S is φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-amenable, then S is amenable. Moreover, φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-ergodicity for a topological semigroup S is introduced and it is proved that under some conditions on S and a Banach space X, φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-amenability and φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-ergodicity of any antirepresntation defined by a right action S on X, are equivalent. A relation between φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-amenability of topological semigroups and the existence of a common fixed point is investigated and by this relation, Hahn-Banach property of topological semigroups in the sense of φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-amenability defined and studied.
