This paper is devoted to the obtaining of the exact solutions in the framework of modifiedf(R)-gravity's rainbow(UV completion of General Relativity)/f(R)-gravity and the study of their thermodynamic behavior. In thef(R)-gravity's rainbow, we have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form (we should note that scalar fieldΨ(x)couples to gravity minimally or non-minimally with coupling constantλand the gravity representation which is coupled with the scalar field isf(R)not justR). The most important result derived from this study is that gravity's rainbow increases the height of this potential in the exterior region of the event horizon. We also figure out the effect of the coupling constantλand thef(R)parameterηon the height of the potential barrier and the other thermodynamical quantities like temperature and heat capacity. We have shown that the effect off(R)-gravity on the temperature adds a coefficient of 1/2 to the cosmological constant. We study the quasinormal modes (QNMs) of 3D black holes in thef(R)-gravity's rainbow. For this purpose, we use the WKB approximation method upto third order corrections. We have shown the perturbations decay in corresponding diagrams when the coupling constantλand rainbow functiong(ε)change. We also obtain two different solutions when Ricci scalar is constant. The first one is for a charged slowly rotating toroidal black holes and the second one is for a higher dimensional toroidal black holes inspired by non-commutative geometry. The source for the second one is given by a fluid-type matter with the Gaussian-distribution smeared mass density. In Starobinsky model we show that the charged slowly rotating solution is ill-defined for s=1 value.
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