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In this paper, we have investigated Tsallis holographic dark energy (infrared cutoff is the Hubble radius) in homogeneous and anisotropic Bianchi type- III  Universe within the framework of Saez–Ballester scalar–tensor theory of gravitation. We have constructed non-interaction and interaction dark energy models by solving the Saez–Ballester field equations. To solve the field equations, we assume a relationship between the metric potentials of the model. We developed the various cosmological parameters (namely deceleration parameter  q , equation of state parameter    $$\omega _t$$  ω t , squared sound speed    $$v_s^2$$  v s 2 , om-diagnostic parameter  Om ( z ) and scalar field    $$\phi $$  ϕ ) and well-known cosmological planes (namely    $$\omega _t-\omega _t^{'}$$  ω t - ω t ′ plane, where    $$'$$  ′ denotes derivative with respect to  ln ( a ) and statefinders (   $$r-s$$  r - s ) plane) and analyzed their behavior through graphical representation for our both the models. It is also, quite interesting to mention here that the obtained results are coincide with the modern observational data.

Bianchi type-III Tsallis holographic dark energy model in Saez–Ballester theory of gravitation