Given an automorphism $$\phi :\Gamma \rightarrow \Gamma $$ ϕ : Γ → Γ of a group, one has a left action of $$\Gamma $$ Γ on itself defined as $$g.x=gx\phi (g^{-1})$$ g . x = g x ϕ ( g - 1 ) . The orbits of this action are called the Reidemeister classes or $$\phi $$ ϕ -twisted conjugacy classes. We denote by $$R(\phi )\in {\mathbb {N}}\cup \{\infty \}$$ R ( ϕ ) ∈ N ∪ { ∞ } the Reidemeister number of $$\phi $$ ϕ , namely, the cardinality of the orbit space $${\mathcal {R}}(\phi )$$ R ( ϕ ) if it is finite and $$R(\phi )=\infty $$ R ( ϕ ) = ∞ if $${\mathcal {R}}(\phi )$$ R ( ϕ ) is infinite. The group $$\Gamma $$ Γ is said to have the $$R_\infty $$ R ∞ -property if $$R(\phi )=\infty $$ R ( ϕ ) = ∞ for all automorphisms $$\phi \in {\text {Aut}}(\Gamma )$$ ϕ ∈ Aut ( Γ ) . We show that the generalized Thompson group T(r, A, P) has the $$R_\infty $$ R ∞ -property when the slope group $$P\subset {\mathbb {R}}^\times _{>0}$$ P ⊂ R > 0 × is not cyclic.