The topology of black hole thermodynamics is a fascinating area of study that explores the connections between thermodynamic properties and topological features of black holes. This paper has led to several significant findings We successfully derive the field equations for FREulerHeisenberg theory providing a framework for studying the interplay between modified gravity and nonlinear electromagnetic effects. We obtain an analytical solution for a static spherically symmetric energydependent black hole with constant scalar curvature. Also our analysis of black holes in FREulerHeisenberg gravitys Rainbow reveals significant insights into their topological properties. We identified the total topological charges by examining the normalized field lines along various free parameters. Our findings indicate that the parameters  R0  and  fepsilon  gepsilon  influence the topological charges. These results are comprehensively summarized in Table I. Additionally a general overview of Tables II III and IV related to the photon sphere of the mentioned black hole reveals that with an increase in fvarepsilon the permissible range of negative lambda in the first case gradually transitions into a nonpermissible region in the third case. On the other hand it is known that the QED parameter which measures the strength of nonlinear effects can be either positive or negative. A positive QED parameter reduces the electric field near the horizon and increases the black holes mass whereas a negative QED parameter increases the electric field and decreases the mass. According to the two statements above it can be concluded that the increase in fvarepsilon actually decreases the strength of the electric field near the horizon and strengthens the effects of gravity.
['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']
Thermodynamic topology of Black Holes in FREulerHeisenberg gravitys Rainbow
{'No Header': '.', "Thermodynamic topology of Black Holes in F ( R ) -Euler-Heisenberg gravity's Rainbow": "Yassine Sekhmani, 1, ∗ Saeed Noori Gashti, 2, † Mohammad Ali S. Afshar, 3, ‡ Mohammad Reza Alipour, 2, § Jafar Sadeghi, 3, ¶ and Javlon Rayimbaev 4, 5, 6, 7,  \n∗∗  \n1 D'epartement de Physique, Equipe des Sciences de la mati'ere et du rayonnement, ESMaR, Facult'e des Sciences, Universit'e Mohammed V de Rabat, Rabat, Morocco 2 School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran 3 Department of Physics, Faculty of Basic Sciences, University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran 4 Institute of Fundamental and Applied Research, National Research University TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan 5 University of Tashkent for Applied Sciences, Str. Gavhar 1, Tashkent 100149, Uzbekistan Shahrisabz State Pedagogical Institute, Shahrisabz Str. 10, Shahrisabz 181301, Uzbekistan  \n6 7 Tashkent State Technical University, Tashkent 100095, Uzbekistan  \nThe topology of black hole thermodynamics is a fascinating area of study that explores the connections between thermodynamic properties and topological features of black holes. It often involves analyzing critical points in the phase diagrams of black holes and assigning topological charges to these points. One significant approach is based on Duan's topological current φ -mapping theory, which introduces the concept of topological charges to critical points in black hole thermodynamics. This paper has led to several significant findings: We successfully derive the field equations for F ( R ) -Euler-Heisenberg theory, providing a framework for studying the interplay between modified gravity and non-linear electromagnetic effects. We obtain an analytical solution for a static, spherically symmetric, energy-dependent black hole with constant scalar curvature. Also, our analysis of black holes in F(R)-Euler-Heisenberg gravity's Rainbow reveals significant insights into their topological properties. We identified the total topological charges by examining the normalized field lines along various free parameters. Our findings indicate that the parameters ( R 0 ) and ( f /epsilon1 = g /epsilon1 ) influence the topological charges. These results are comprehensively summarized in Table I. Additionally, a general overview of Tables II, III, and IV related to the photon sphere of the mentioned black hole reveals that with an increase in f ε , the permissible range of negative λ in the first case gradually transitions into a non-permissible region in the third case. On the other hand, it is known that the QED parameter, which measures the strength of nonlinear effects, can be either positive or negative. A positive QED parameter reduces the electric field near the horizon and increases the black hole's mass, whereas a negative QED parameter increases the electric field and decreases the mass. According to the two statements above, it can be concluded that the increase in f ε actually decreases the strength of the electric field near the horizon and strengthens the effects of gravity.", 'I. INTRODUCTION': "The greatest cosmological challenges this century are early-time inflation and late-time acceleration. In precise terms, what does inflation refer to, and what does dark energy consist of? Why and how did the early universe and the late universe expand upwards so very fast? What might explain why the evolution of the universe is similar for small and large curvatures? To  \ntackle these matters, Refs. [1-5] outline and investigate Einstein's modified theories of gravity as an alternative framework. Among the various modified theories of gravity, the F ( R ) theory has aroused a great deal of interest because it offers an explanation for a broad series of observations. For instance, the F ( R ) theory of gravity provides a plausible hypothesis for the existence of dark matte [6, 7] and the universe's rapid expansion [25, 8, 9]. Even more, the F ( R ) theory can be harnessed to forecast cosmic acceleration or early universe inflation [10-12], not to mention massive compact objects [1318]. Another achievement in the context of F ( R ) gravity is the complete description of all the evolutionary epochs of the universe while maintaining consistency with Newtonian and post-Newtonian approximations [19]. Within the theory of F ( R ) gravity, F ( R ) turns out  \nto be an arbitrary function of the scalar curvature R .  \nSince the theory of F ( R ) gravity may describe a number of cosmological and astrophysical phenomena, it is vital to examine and highlight the coupling of the EulerHeisenberg (EH) term as a matter source with F ( R ) gravity's rainbow by adopting the F ( R ) -EHLagrangian. Heisenberg and Euler established the non-linear electrodynamics Lagrangian making use of Dirac's electronpositron theory [20]. Schwinger remodelled this nonperturbative one-loop Lagrangian in the context of quantum electrodynamics (QED) [21] and found that it could identify the phenomenon of vacuum polarisation successfully. Due to the one-loop nonperturbative QED, the EH Lagrangian has drawn ever-increasing scrutiny to the subject of generalised black hole solutions. A wealth of intriguing studies on black holes in the context of EH nonlinear electrodynamics are described in [22-29].  \nIn the search for the unification of general relativity and quantum mechanics, a wide range of challenges have arisen. Nowadays, in spite of a few theories such as string theory [30], loop quantum gravity [36], and space-time foam models [32] that have attempted to tackle this conundrum, the question remains devoid of any phenomenological process. A key reason for unifying gravity with quantum theory is the violation of Lorentz invariance, which we might regard as an indispensable criterion for the elaboration of a quantum theory of gravity. Another distinctive factor among unification theories is that they forecast a maximum energy value of the order at the Planck scale that a particle could attain; i.e., if a particle is qualified by the Standard Model of particle physics, then it should conform to the upper limit of the Planck energy, 5.10 19 eV , which is experimentally backed up by Abraham et al [33]. A further alternative to Lorentz symmetry violation involves, in the ultraviolet limit, modifying the standard energymoment dispersion ratio. This, in fact, has a long history in theories as diverse as discreteness space-time [39], Horava-Lifshitz [34, 35], doubly special relativity [40, 41], and loop quantum gravity [36, 37]. In the framework of doubly special relativity, the dispersion relation works out as E 2 f 2 /epsilon1 -p 2 g 2 /epsilon1 = m 2 where f /epsilon1 and g /epsilon1 are functions of /epsilon1 = E / E P , E standing for the energy of the particle used to analyse space-time and E P for Planck's energy. The functions f /epsilon1 and g /epsilon1 are commonly thought of as rainbow functions and are designed for phenomenological purposes [42]. On the other hand, the standard energy dispersion relation gets recurved in the limit lim /epsilon1 → 0 f /epsilon1 = lim /epsilon1 → 0 g /epsilon1 = 1, otherwise known as the infrared limit. As a matter of fact, Linear Lorentz transformations break the invariance of the theory [43]; nevertheless, non-linear Lorentz transformations are those  \nthat keep the theory invariant. Moreover, as well as the speed of light being a constant, the Planck energy is also a constant, and it remains impossible for a particle to rise beyond this limit.  \nA generalisation of doubly special relativity to what is known as rainbow gravity has been suggested by Magueijo and Smolin [44], where spacetime is mapped to a family of specified parameters in the metric that is parametrised by /epsilon1 . Accordingly, the spacetime geometry is contingent on the energy of the particle testing it, causing a rainbow of the metric. Meanwhile, a number of studies have incorporated Magueijo and Smolin's theory into the understanding of black holes [45-59] and into modifying theories of gravity [60, 61]. A worthy concrete illustration investigation is concerned with the work of M. Momennia et al. [62] exploiting a non-linear electrodynamics (NED) field in the context of the rainbow gravity, resulting in the validity of the first law of thermodynamics in the presence of rainbow functions. Therefore, the rainbow gravity framework has provided the avenue for the consideration of a large number of investigations, such as the derivation considering the dilatonic gravity of an exact black hole solution minimally coupled to the Born-Infeld term with an energy-dependent Liouville-type potential [63]. Similarly, incorporating the quadratic term of the GaussBonnet gravity in the framework of rainbow gravity leads to modifying the appropriate constraints to work out non-singular universes [64]. It is worth noting that the study of thermodynamic properties is considered the main and impacting process regarding rainbow gravity. Thus, since the energy scale depends on the spacetime metric, entropy and temperature also have the same dependence [65]. To gain a topological perspective in thermodynamics, one effective approach is to use Duans topological current φ mapping theory. Wei et al. introduced two distinct methods to study topological thermodynamics based on temperature and the generalized free energy function. The first method involves analyzing the temperature function by eliminating pressure and utilizing the auxiliary and topological parameter 1/ sin θ . The potential is then constructed based on these assumptions. For further study, see [6669]. The second method assumes that black holes can be considered defects in the thermodynamic parameter space. Their solutions are investigated using the generalized off-shell free energy. In this context, the stability and instability of the obtained black hole solutions are determined by positive and negative winding numbers, respectively. Additionally, the properties of a field configuration can be deduced from the zero points of the field in space [70-73]. Wei, Liu, and B. Mann asserted  \nthat for different branches of a black hole at an arbitrary temperature, the topological charge number, the sum of the winding numbers, is a universal number independent of the black hole parameters. In the range of small and large black holes, the topological charge number depends solely on the thermodynamic asymptotic behavior of the black hole temperature. Consequently, black holes can be categorized into three different groups based on the number of topological charges [70]. Following the introduction of this second method, various black holes were studied to investigate topological charges. C. H. Liu and J. Wang demonstrated that the topological number remains constant, and the charged GB black holes in the AdS space fall into the same category as the RN-AdS, sharing the same topological numbers. However, their results indicated that charge dependence is not the only effective factor [74]. Y. Du and Xi. Zhang examined the rotating charged BTZ black hole model and found that there are only two topological classes for BTZ space-time. [75]. Using the same method, Wu investigated the structure of charged Lorentzian Taub-NUT space-times and neutral Lorentzian NUT-charged space-times in 4 dimensions in two separate articles. [76]. In the second work, Wu found that the presence of the NUT charge as a parameter did not affect the structures in symptotically flat space-time but had an effect on asymptotically local AdS space-time [77]. For further study, you can see [78-95]. Building upon the above concepts, we organize the article as follows: In Section II, we provide the field equations in F(R)-Euler-Heisenberg theory. In Section III, we obtain the black hole solutions in F(R)-Euler-Heisenberg gravity's rainbow. In Sections IV and V, we study the thermodynamics, thermodynamic topology, and photon sphere of the mentioned black holes. We obtain the thermodynamic topology and topological classification of black holes in F(R)-Euler-Heisenberg gravity's rainbow and its photon sphere in some cases in Section VI. Finally, we present the conclusion of our paper in Section VII.", 'II. FIELD EQUATIONS IN F ( R ) -EULER-HEISENBERG THEORY': "In this section, we aim to highlight the coupling of the EH term as a matter source with the F ( R ) gravity. To this end, we consider the F ( R ) -EH action  \nI F ( R ) = 1 16 π ∫ M d 4 x √ -g ( F ( R ) -L ( S , P ) ) (1)  \nwhere F ( R ) = R + f ( R ) , in which R and f ( R ) , are respectively, the Ricci scalar and a general function of the Ricci scalar, g = det ( g µν ) is the determinant of the metric tensor g µν , and L ( S , Q ) is dedicated to being the EH Lagrangian. Throughout this study, we adopt the Newtonian gravitational constant and the speed of light as being equal to 1, i.e., G = c = 1. In practical terms, the EH Lagrangian can be described by considering the following expression [20]:  \nL ( S , Q ) = -S + λ 2 S 2 + 7 λ 8 Q 2 (2)  \nwhere λ = 8 α 2 /45 m 4 e is the EH parameter by which the intensity of the NLED contribution is regulated; α is the fine structure constant; and me is the mass of the electron; hence, the EH parameter ( λ ) is of the order of α / E 2 c . In turn, S and Q are, respectively, true scalars and pseudo-scalars, with the following definitions:  \nS = F 4 , Q = ˜ F 4 (3)  \nwhere F = F µν F µν stands for the Maxwell invariant together with F µν = ∂µ A ν -∂ν A µ is the electromagnetic field strength of which A µ is the gauge potential. Furthermore, the invariant ˜ F can be defined in terms of ˜ F = F µν ˜ F µν , where ˜ F µν = 1/2 /epsilon1 ρλ µν F ρλ . Notably, the completely antisymmetric tensor /epsilon1µνρσ , satisfies /epsilon1µνρσ/epsilon1 µνρσ = -4. It is worth pointing out that the λ = 0 criterion abolishes the non-linear electromagnetic character of the EH theory so that the linear electromagnetic field of the Maxwell theory arises ( L ( S ) = -S ) .  \nAs far as NLEDs are concerned, though, there are two potential frameworks: one is the usual framework (the F -frame) in terms of the electromagnetic field tensor F µν . The other frame is the P frame, with the P µν tensor as the principal field of the definition  \nP µν = -( LS F µν + ˜ F µν LP ) (4)  \nwhere LX = ∂ L ∂ X with X = ( S , Q ) . In the realm of the EH theory, P µν can be defined by  \nP µν = ( 1 -λ F ) F µν -˜ F µν 7 λ 4 Q (5)  \nwhich stands for a class of electric/magnetic fields, namely the electric induction D and the magnetic field H . In connection to the condensed matter fields, the EH term, according to Eq. (18), involves a duality between the vectors set such that D , H and the magnetic intensity B and the electric field E .  \nTo better grasp the EH model by means of the P frame, it is quite worthwhile to consider the two independent  \ninvariants P and O assigned to the P frame, which are given by  \nP = -1 4 P µν P µν , O = -1 4 P µν ∗ P µν (6)  \nwhere ˜ P µν = 1 2 √ -g /epsilon1µνρσ P σρ . In particular, the Legendre transformation of the relevant Lagrangian ' L is exploited to provide a structural function H expressed as follows:  \nH ( P , O ) = -1 2 P µν F µν -L (7)  \nwhere, regardless of the second and higher order terms in λ , this structural function can be represented as follows:  \nH ( P , O ) = P -λ 2 P 2 -7 λ 8 O 2 . (8)  \nThe equations of motion of the F ( R ) -EH theory of gravity can be obtained, leading to  \n8 π T µν = R µν ( 1 + f R ) -g µν F ( R ) 2 +( g µν ∇ 2 -∇ µ ∇ ν ) f R (9)  \n∇ µ P µν = 0 (10)  \nwhere f R = d f ( R ) d R and T µν is the energy-momentum tensor for the EH theory in the P setting, given by  \nT µν = 1 4 π ( ( 1 -λ P ) P β µ P νβ + g µν ( P -3 2 λ P 2 -7 λ 8 O 2 ) ) . (11)  \nNext, we present the spherically symmetric static solution for the F ( R ) -EH gravity's rainbow.", "III. BLACK HOLE SOLUTIONS IN F ( R ) -EULER-HEISENBERG GRAVITY'S RAINBOW": "Within the scope of this study, we consider a spherically, static, four-dimensional, energy-dependent spacetime, in which we adopt the procedure described in Refs. [44, 101], so that  \nh ( /epsilon1 ) = η µν e µ ( /epsilon1 ) ⊗ e ν ( /epsilon1 ) (12)  \nand  \ne 0 ( /epsilon1 ) = 1 f /epsilon1 ˜ e 0 , e i ( /epsilon1 ) = 1 g /epsilon1 ˜ e i (13)  \nwhere the quantities in tilde (i.e. ˜ e 0 and ˜ e i ) stand for energy-independent frame fields. Based upon the foregoing assumptions, we in fact may generate black holes  \nin F ( R ) -EH gravity's rainbow by considering the following static, spherically symmetric, energy-dependent, four-dimensional spacetime  \nd s 2 = -F ( r ) f 2 /epsilon1 d t 2 + 1 g 2 /epsilon1 ( d r 2 F ( r ) + r 2 ( d θ 2 + sin 2 θ d φ 2 ) ) (14)  \nwhere F ( r ) , f /epsilon1 and g /epsilon1 is the metric function , and rainbow functions.  \nPractically speaking, the field equations underlying the context of F ( R ) gravity with the EH matter fields (9) seem to be difficult. This means that accurate analytical solutions are proving tricky to pin down. To overcome this issue, one should consider the traceless energy-momentum tensor for the EH matter field, and then one can obtain an analytical solution from F ( R ) gravity coupled to a nonlinear electrodynamics field. Thus, to be able to derive the solution of a black hole with constant curvature in the F ( R ) theory of gravity as coupled to the EH matter field, it suffices for the trace of the stress-energy tensor T µν to hold to zero [102, 103]. Based on the hypothesis of constant scalar curvature R = R 0 = constant [104], the trace of equation (9) takes the following terms:  \nR 0 ( 1 + f R 0 ) -2 ( R 0 + f ( 0 )) = 0 (15)  \nwhere f R 0 = f R | R = R 0 . The constant scalar curvature is accordingly drawn from the equation (15) as follows:  \nR 0 = 2 f ( R 0 ) f R 0 -1 . (16)  \nMaking use of Eq. (16) into Eq. (9), the field equations of the F ( R ) -EH theory could be established in the right way as  \nR µν ( 1 + f R 0 ) -g µν 4 R 0 ( 1 + f R 0 ) = 8 π T µν . (17)  \nBywayofobservation, the parameter f R 0 implies a gravitational redefinition concerning general relativity; in particular, imposing the condition f R 0 = 0 leads to a redefinition of the general relativity. Thus, the field equation in the F ( R ) -EHtheory (17) reduces to the EH theory of gravity.  \nTurning to the electromagnetic field, we have to treat only the electric field source of the EH theory so that the electromagnetic tensor is explicitly derived as  \nP µν = q r 2 δ 0 [ µ δ 1 ν ] (18)  \nin which the electromagnetic invariants are given by  \nP = q 2 2 r 4 , O = 0. (19)  \nwhere q represents an integration constant that is associated with the electric charge.  \nAt this stage, the field equations for F ( R ) gravity with EH matter fields (9), in conjunction with the consideration of the metric function (14) and the tensor of the electric component (18), are explicitly expressed by the following differential equations:  \neqtt = eqrr = rF ' ( r ) + F ( r ) + r 2 R 0 4 g 2 /epsilon1 -1 -f 2 /epsilon1 ( 1 + f R 0 ) ( λ q 2 4 r 6 -q 2 r 2 ) , (20)  \neq θθ = eq φφ = rF '' ( r ) + 2 F ' ( r ) + rR 0 2 g 2 /epsilon1 + f 2 /epsilon1 ( 1 + f R 0 ) ( 3 4 λ q 4 r 7 -q 2 r 3 ) (21)  \nwhere eqtt , eqrr , eq θθ and eq φφ , are, respectively, the tt , rr , θθ and φφ components of the field equations (9). By implementing the above differential equation set, a definite solution to the constant scalar curvature ( R = R 0 = constant ) can be provided. Following a close examination and further calculations, the metric function can now be expressed in the most precise form.  \nF ( r ) = 1 -m r -R 0 r 2 12 g 2 /epsilon1 + f 2 /epsilon1 ( 1 + f R 0 ) ( q 2 r 2 -λ q 4 20 r 6 ) (22)  \n/negationslash  \nwhere m 0 is an integration constant correlated with the black hole's geometric mass. It is worth highlighting that the obtained black hole solution (22) is exactly in conformity with the field equations (9). Therefore, to carry out a physical analysis based on the black hole solution (22), one can ensure a constraint on the parameter f R 0 such that f R 0 = 1. On the other hand, the consideration of the particular partial space, namely ( f R 0 = 0, R 0 = 4 Λ , f 2 /epsilon1 = g 2 /epsilon1 = 1 and λ = 0), recovers the so-called Reissner-Nordstrom-(A)dS black hole defined by  \nF ( r ) = 1 -m r -Λ r 2 3 + q 2 r 2 -λ q 4 20 r 6 . (23)  \nSuch application tools are needed to understand the black hole solution better. In particular, the singularity and uniqueness features can be exploited by using the scalar invariants, i.e., the Ricci scalar, the Ricci square, and the Kretschmann scalar.  \nThe Ricci scalar for the corresponding metric is given by  \nR = R 0 g 2 /epsilon1 + f 2 /epsilon1 λ q 4 ( 1 + f R 0 ) r 8 . (24)  \nThe Ricci squared is given by  \nR µν R µν = 1 4 g 4 /epsilon1 r 16 ( f R 0 + 1 ) 2   5 f 4 /epsilon1 g 4 /epsilon1 λ 2 q 8 -16 f 4 /epsilon1 g 4 /epsilon1 λ q 6 r 4 + 2 f 2 /epsilon1 g 2 /epsilon1 q 4 r 8 × ( 8 f 2 /epsilon1 g 2 /epsilon1 + λ R 0 ( f R 0 + 1 ) ) + r 16 R 2 0 ( f R 0 + 1 ) 2   . (25)  \nOn the other hand, the Kretschmann scalar is formulated according to the metric function (14) and the F ( r ) fucntion (22) as follows:  \nR αβµν R αβµν = 1 150 g 4 r 16 ( f R 0 + 1 ) 2   10 f 2 g 2 λ q 4 r 4 × ( r ( f R 0 + 1 ) ( 168 g 2 m 0 + 5 r 3 R 0 ) -456 f 2 g 2 q 2 ) + 25 r 8 ( -288 f 2 g 4 m 0 q 2 r × ( f R 0 + 1 ) + 336 f 4 g 4 q 4 + r 2 ( f R 0 + 1 ) 2 × ( 72 g 4 m 2 0 + r 6 R 2 0 ) ) + 717 f 4 g 4 λ 2 q 8   . (26)  \nUpon thorough analysis of the expressions (24), (25), and (26), the black hole solution described by the metric (14) is singular for all the allowable values of the parameters black hole system. So investigating the limits for the scalar invariants at r = 0 would result in the following:  \nlim r → 0             R ≈ ∞ R µν R µν ≈ ∞ R αβµν R αβµν ≈ ∞ . (27)  \n By way of observation, the presence of the spacetime singularity is due to the mass ( m 0 ) , the electric charge ( q ) , and the EH parameter ( λ ) in the black hole metric. A useful way of avoiding the singularity resulting from the mass, the charge, and the EH parameters is to outline a procedure with a non-linear charge distribution function similar to Ref. [105]. Throughout this work, we will not be thinking about such a situation and will stick to the metric function (14) for the rest of the analysis. On  \nFIG. 1: F ( r ) versus r for various values of the parameter m 0 .  \n<!-- image -->  \nthe other hand, examining behavior at large distances is a worthwhile adjunct, given that  \nlim r → ∞                 R ≈ R 0 g 2 /epsilon1 R µν R µν ≈ R 2 0 4 g 4 /epsilon1 R αβµν R αβµν ≈ R 2 0 4 g 2 /epsilon1 . (28)  \n which implies that the Ricci scalar, the Ricci squared, and the Kretschmann scalar have a finite term at a large distance. To sum up, these scalars are evidence that our black hole solution is unique, and that the EH parameter does not modify the asymptotical behavior of the spacetime  \nNext, the examination of the metric function allows to explore the set of corresponding real roots. On the other hand, the analysis of the appropriate set of these real roots offers the possibility to extract information on the horizons (inner and outer horizons, and even more the cosmological horizon). Thus, to proceed with this examination, such a numerical approach is needed.  \nSolving the equation B ( r = r h ) = 0 at the event horizon leads to specifying the horizon radii spectrum. For this reason, Fig. 1 represents, either for R 0 < 0 (or AdS space if R 0 = 4 Λ ) or for R 0 > 0 (or dS space if R 0 = 4 Λ ), the variation of the metric function as a function of the space-time variable r graphically showing the proper horizon radii. It is worth remarking that in the case of R 0 > 0, the horizon structure provides at most four horizon radii (one of which refers to the cosmological horizon). In contrast, as we can observe, the case  \nrelated to R 0 < 0 can cover only three possible horizon radii (one of which is adapted to the cosmological horizon). Roughly speaking, the sign of the parameter R 0 plays a crucial role in defining the structure of black hole horizon in the F ( R ) -EH gravity's rainbow.  \nAnalysing the impact of the electrical charge ( q ), the parameter of EH ( λ ), and F ( R ) 's parameters ( f R 0 , and R 0 ) and Rainbow structure ( f /epsilon1 , and g /epsilon1 ) on the structure of the black hole horizon in the F ( R ) -EH gravity's rainbow is also necessary. To this end, Fig. 2 depicts the behaviour of the metric function against the spacetime variable r . It is observed that the appropriate horizon structure provides at most three horizon radii (one of which corresponds to the cosmological horizon) for the case where R 0 < 0. Thus, each parameter variation affects the horizon structure in such a way that we can point out that at a critical value of the EH parameter, i.e., λ crit = 0.145, there are two horizon radii, which are a root (in the extreme case) and the cosmological horizon. Further, at the range λ crit < λ , the horizon structure admits three horizon radii, namely the inner horizon, the outer horizon, and the cosmological horizon. Whereas for λ crit > λ , the horizon structure merely carries the cosmological horizon. Similarly, a graphical inspection of the electrical charge parameter ( q ) may imply a particular pattern for the horizon structure. So, we can observe that for a certain critical value, such as q crit = 0.95, the horizon structure may entail two horizon radii, the inner horizon being the smallest among the three horizons and the other being formed by a matching between the outer horizon and the cosmological horizon. Furthermore, for q crit > q , the horizon structure involves  \nm  \n=  \n2,  \nR  \n=-  \n1,  \nfR  \n=  \n0.1,  \nFIG. 2: F ( r ) versus r for various values of the parameter space ( R 0 , f R 0 , q , λ , f /epsilon1 , g /epsilon1 ) .  \n<!-- image -->  \nλ  \n=  \n0.085, q  \n=  \n0.85,  \ng  \n=  \n1.2  \nonly the inner horizon, whereas for q crit < q , the situation consists of three horizon radii. As far as the pa-  \nrameters of F ( R ) are concerned, namely f R 0 and R 0 , the horizon structure, for instance, in terms of R 0 entails a critical value R crit 0 = -9.45 at which it exhibits two types of horizon, the inner horizon and the other being formed by a correlation between the outer horizon and the cosmological horizon. In addition, for R crit 0 > R 0 , the black hole horizon carries three horizon radii, while in the case of R crit 0 < R 0 , the horizon structure has only one horizon, which stands for the outer. On the other hand, it is notable that for a critical value of the parameter f R 0 such that f crit R 0 = 0.215, the black hole admits in particular the extreme horizon and the cosmological horizon. For f R crit 0 > f R 0 , the situation turns out to be different since the black hole's horizon entails only the cosmological horizon. Moreover, in the case f R crit 0 < f R 0 , the horizon structure exhibits three horizon radii, i.e., all the possible horizons of the black hole. Finally, it is necessary to explore the characteristic of the horizon structure concerning the variation of the set of rainbow gravity functions, such as ( f /epsilon1 , g /epsilon1 ). This variation can be appreciated on the plot, in which, at a critical value of g /epsilon1 as g crit /epsilon1 = 0.39, the horizon structure possesses the inner horizon and one that is correlated between the outer horizon and the cosmological horizon. Other scenarios can be considered to clearly assess the impact of the Rainbow function such as g /epsilon1 on the horizon, from which at g /epsilon1 > g crit /epsilon1 , the black hole admits three horizons, whereas for g /epsilon1 < g crit /epsilon1 , the horizon involves only the inner horizon. On the other hand, we observe that the horizon structure comprises the inner horizon and a correlated horizon between the outer horizon and the cosmological horizon in the case where f crit /epsilon1 = 1.21, i.e., above this threshold, the black hole admits only the inner horizon, and below this threshold, the black hole presents three horizon radii.", 'IV. THERMODYNAMICS': "The best way to approach thermodynamically the black hole solutions in F ( R ) gravity's rainbow with EH matter source is to consider the first law of thermodynamics. To begin with, inspection at the event horizon r = r h of the metric solution (22) leads to providing the mass term of the black hole as follows:  \nm = r h -r 3 h R 0 12 g 2 /epsilon1 -f 2 /epsilon1 ( f R 0 + 1 ) ( λ q 4 20 r 5 h + q 2 r ) . (29)  \nNext, to ascertain the Hawking temperature, one has to consider primarily the surface gravity [106], which is  \nsupplied out of  \nκ = ( -1 2 ∇ µξν ∇ µ ξ ν ) 1/2 ∣ ∣ ∣ ∣ r = r h = F ' ( r ) g /epsilon1 f /epsilon1 ∣ ∣ ∣ ∣ r = r h (30)  \n∣ ∣ with ξ µ = ∂ / ∂ t is a Killing vector. Achieving a correct expression of the surface gravity ( κ ) is carried out by exploiting the metric function (22) and by injecting the mass (29) into Eq. (30). The surface gravity of a black hole is therefore expressed by  \nκ = f /epsilon1 g /epsilon1 8 ( f R 0 + 1 ) r 7 h ( λ q 4 -4 q 2 r 4 h ) -R 0 r h 8 f /epsilon1 g /epsilon1 + g /epsilon1 2 f /epsilon1 r h , (31)  \nas well as the formula T = κ /2 π stands for the Hawking temperature, which could be expressed by an appropriate term as follows:  \nT = f /epsilon1 g /epsilon1 16 π ( f R 0 + 1 ) r 7 h ( λ q 4 -4 q 2 r 4 h ) -R 0 r h 16 π f /epsilon1 g /epsilon1 + g /epsilon1 4 π f /epsilon1 r h . (32)  \nIt is noteworthy that all the parameters set in the black hole system, namely, R 0 , f R 0 , q , λ , g /epsilon1 and f /epsilon1 perfectly affected the behavior of the Hawking temperature.  \nTo achieve an accurate description of the Hawking temperature behavior, it is merely instructive to study the asymptotic and high-energy limits of the temperature. In this respect, the high-energy limit can be exemplified as follows  \nlim r h → 0 T ∝ f /epsilon1 g /epsilon1 λ q 4 16 π ( f R 0 + 1 ) r 7 h (33)  \nwhich proves the extent to which the parameters of the matter sector ( λ , q ) as well as the rainbow functions ( f /epsilon1 , g /epsilon1 ) influence the high-energy behavior of the Hawking temperature. Accordingly, the Hawking temperature is definitely positive for at the high-energy limit.  \nOn the other hand, the asymptotic limit of temperature remains just a dependence of R 0 , such that  \nlim r h → ∞ T ∝ -R 0 r h 16 π f /epsilon1 g /epsilon1 (34)  \nwhich implies that the Hawking temperature sets to be positive (negative) whenever R 0 < 0 ( R 0 > 0).  \nBy using Gauss's law, the electric charge of the black hole per unit volume, V , can be ascertained by the following formula  \nQ = ˜ Q V = qf /epsilon1 4 π g /epsilon1 . (35)  \nThe EH Φ electromagnetic potential corresponding to the black hole within the framework of the F(R)-EH gravity's rainbow is as follows  \nΦ = ∫ ∞ r h d rPtr H P ( P , 0 ) = ∫ ∞ r h d r q r 2 ( 1 -λ q 2 2 r 4 ) = q r h ( 1 -λ q 2 10 r 4 h ) . (36)  \nBy applying a modified area law in the F ( R ) theory of gravity, we can extrapolate the corresponding entropy of black holes such as  \nS = A ( 1 + f R 0 ) 4 (37)  \nwith A being the horizon area as  \nA = ∫ 2 π 0 ∫ π 0 √ g θθ g φφ ∣ ∣ ∣ ∣ r = r h = r 2 g 2 /epsilon1 ∣ ∣ ∣ ∣ r = r h = r 2 h g 2 /epsilon1 (38)  \n∣ ∣ So now we can work out the entropy of EH black holes in F ( R ) gravity's rainbow by substituting Eq. (38) into Eq. (37), thus yielding  \nS = ˜ S V = ( 1 + f R 0 ) r 2 h 4 g 2 /epsilon1 (39)  \nwhich states that the law of areas cannot be held for black hole solutions in gravity R + f ( R ) .  \nExploring the total mass of black holes in F ( R ) -EH theory would be achievable by looking at the AshtekarMagnon-Das (AMD) approach [107, 108], which can be expressed as follows  \nM = ˜ M V = m ( 1 + f R 0 ) 8 π g /epsilon1 f /epsilon1 . (40)  \nThis allows us, after substituting mass (29) in Eq. (40), to obtain the total mass as follows:  \nM = ( f R 0 + 1 ) ( 12 g 2 /epsilon1 r h -R 0 r 3 h ) 96 π f /epsilon1 g 3 /epsilon1 + f /epsilon1 ( 20 q 2 r 4 h -λ q 4 ) 160 π g /epsilon1 r 5 h (41)  \nAfterwards, inspection of the total mass at its highenergy limit can reveal the form  \nlim r h → 0 M ∝ -f /epsilon1 λ q 4 160 π g /epsilon1 r 5 h (42)  \nwhere a consistent dependency is shown only with parameters ( q , λ , g /epsilon1 and f /epsilon1 ). Thus, at the high-energy limit, it is striking that the total mass of small black holes is always negative.  \nThe asymptotic limit, on the other hand, of the total mass is found to be as follows:  \nlim r h → ∞ M ∝ -( f R 0 + 1 ) R 0 r 3 h 96 π f /epsilon1 g 3 /epsilon1 (43)  \nwhere the positive (negative) branch of M is consistently referred to as R 0 < 0 ( R 0 > 0).  \nAt this stage, we may evaluate the first law of thermodynamics. Accordingly, the conserved and specific thermodynamic quantities in Eqs. (41), (32), (36) and (37) satisfy the first law of thermodynamics in the following format.  \nd M = T d S + Φ d Q (44)  \nwhere T = ( ∂ M ∂ S ) Q , and Φ = ( ∂ M ∂ Q ) S , respectively, are in accordance with the relations achieved in the Eqs. (32) and (36).", 'V. THERMODYNAMICTOPOLOGY AND PHOTON SPHERE': "Thermodynamic topology involves studying the thermodynamic properties of black holes using topological methods. It helps understand the critical points and phase transitions of black holes by analyzing their topological charges and numbers. A photon sphere is a region where gravity is strong enough that photons (light particles) are forced to travel in orbits. For a black hole, this sphere lies just outside the event horizon. The existence and properties of photon spheres can be studied using topological methods, which assign topological charges to these spheres. These charges help in understanding the stability and structure of the photon sphere. In recent studies, researchers have explored the relationship between thermodynamic topology and photon spheres in various black hole models, including those with hyperscaling violation. They have found that different topological charges correspond to different physical configurations, such as black holes and naked singularities[81]. Here, we want to study the Thermodynamic topology of Black Holes in F ( R ) -EulerHeisenberg gravity's Rainbow. so we follow some route that is explained in detail in the following section", 'VI. THERMODYNAMICSTOPOLOGY OF BLACK HOLES': "The primary concept associated with defects is the topological charge. To examine the thermodynamic topology of a black hole, we calculate the topological  \ncharge and utilize it to determine the topological classes. We employ Duan's φ mapping technique to compute the topological charge. To introduce the thermodynamic properties of black holes, various quantities are utilized. For example, two distinct variables, such as mass and temperature, can describe the generalized free energy[109]. Given the relationship between mass and energy in black holes, we can reformulate our generalized free energy function as a standard thermodynamic function in the following form,  \nF = M -S τ , (45)  \nwhere, τ represents the Euclidean time period, and T (the inverse of τ ) denotes the temperature of the ensemble. The generalized free energy is on-shell only when τ = τ H = 1 T H . From the expression for the off-shell free energy of a black hole, a vector field is constructed as follows,  \nφ = ( φ r H , φ Θ ) = ( ∂ F ∂ r H , -cot Θ csc Θ ) , (46)  \nwhere φ Θ is divergent, the vector direction points outward when Θ = 0 or π . For r H and Θ , the ranges are 0 ≤ r H ≤ ∞ and 0 ≤ Θ ≤ π , respectively. A topological current can be defined using Duan's φ -mapping topological current theory as follows,  \nj µ = 1 2 π ε µνρ ε ab ∂ν n a ∂ρ n b , µ , ν , ρ = 0, 1, 2 (47)  \nTo determine the topological charge, we first identify the unit vector n , where n = ( n 1 , n 2 ) . We have n 1 = φ r H ‖ φ ‖ and n 2 = φ Θ ‖ φ ‖ . According to Noether's theorem, the resulting topological currents are conserved viz ∂µ j µ = 0,. It can be demonstrated that the current j µ is non-zero only at the zero points of the vector field by utilizing the following equation,  \nj µ = δ 2 ( φ ) J µ ( φ x ) , (48)  \nwhere we utilize the following properties of the Jacobi tensor,  \nε ab J µ ( φ x ) = ε µνρ ∂νφ a ∂ρφ b . (49)  \nThe Jacobi vector reduces to the standard Jacobi when µ = 0, as demonstrated by J 0 ( φ x ) = ∂ ( φ 1 , φ 2 ) ∂ ( x 1 , x 2 ) . Equation (48) shows that j µ is non-zero only when φ = 0. Through some calculations, we can express the topological number or total charge W in the following form,  \nW = ∫ Σ j 0 d 2 x = Σ n i = 1 β i η i = Σ n i = 1 ω i , (50)  \nwhere β i is the positive Hopf index, counting the loops of the vector φ a in the φ space when x µ is near the zero point z i . Meanwhile, η i = sign ( j 0 ( φ / x ) z i ) = ± 1. The quantity ω i represents the winding number for the i -th zero point of φ in Σ . Note that the winding number is independent of the shape of the region where the calculation occurs. The value of the winding number is directly associated with black hole stability. The topological charge W can be determined by summing the winding numbers calculated along each contour surrounding the zero points,  \nW = ∑ i ω i . (51)  \nIt is important to note that if the parameter region lacks zero points, the total topological charge is considered to be zero. In the following two subsections, we will explore the thermodynamic topology of black holes in Euler-Heisenberg F(R)-Rainbow gravity", "A. Topological classification of black holes in F ( R ) -Euler-Heisenberg gravity's Rainbow": "Based on equations (45), we derive the generalized Helmholtz free energy for black holes within the framework of Euler-Heisenberg F(R)-Rainbow gravity  \nF = -120 ( c + 1 ) f /epsilon1 g /epsilon1 r 7 H + 5 ( c + 1 ) r 6 H τ ( r 2 H R 0 -12 g 2 /epsilon1 ) + 3 f 2 /epsilon1 g 2 /epsilon1 q 2 τ ( λ q 2 -20 r 4 H ) / 480 π f /epsilon1 g 3 /epsilon1 r 5 H τ (52)  \nFrom the discussion in the previous section, the form of the function ( φ r , φ Θ ) is determined as follows,  \nφ r H = -16 ( c + 1 ) f /epsilon1 g /epsilon1 r 7 H +( c + 1 ) r 6 H τ ( 4 g 2 /epsilon1 -r 2 H R 0 ) + f 2 /epsilon1 g 2 /epsilon1 q 2 τ ( λ q 2 -4 r 4 H ) / 32 π f /epsilon1 g 3 /epsilon1 r 6 H τ φ Θ = -cot ( Θ ) sin ( Θ ) (53)  \nThe unit vectors n 1 and n 2 are computed using equation (53). Next, we calculate the zero points of the φ r H component by solving the equation φ r H = 0 and derive an expression for τ as follows,  \nτ = [ 16 ( c + 1 ) f /epsilon1 g /epsilon1 r 7 H ] / [ 4 cg 2 /epsilon1 r 6 H -cr 8 H R 0 + f 2 /epsilon1 g 2 /epsilon1 λ q 4 -4 f 2 /epsilon1 g 2 /epsilon1 q 2 r 4 H + 4 g 2 /epsilon1 r 6 H -r 8 H R 0 ] (54)  \nWe analyze Figure (3), which pertains to the structure of black holes in Euler-Heisenberg F(R)-Rainbow gravity. The figures display the normalized field lines. As depicted in the figures, some zero points represent the model's topological charges with various free parameters. They are proportional to the winding number and are situated inside the blue contour at the coordinates ( r H , Θ ) . We considered free parameters q = 0.5, λ = 0.5, c = 0.01, R 0 = + 0.001 to draw these contours. In Figure (3,left), we plotted the curve for f /epsilon1 = g /epsilon1 = 1.2. As shown in Figure (3,left), three blue loops represent the topological numbers ω =(-1, +1, -1) and the total topological number W = -1. Also, in Figures (3,middle) and (3,right), we plotted the curve for f /epsilon1 = g /epsilon1 = 1.2, 1.4 respectively. (3,middle) represents the topological numbers -1, + 1 with the total topological number W = 0 and Figure (3,right) represents the one topological number -1, so that the total topological number is W = -1. Without loss of generality, we analyzed the topological properties of these black holes in this paper by choosing τ for black holes in Euler-Heisenberg F(R)-Rainbow gravity. The content discusses the stability of the black holes by examining the winding numbers and specific heat capacity. We have plotted Figures (4,top;left), (4,top; middle and righr) and (4,bottom; left and right) for the mentioned model with respect to free parameters q = 0.5, λ = 0.5, c = 0.01, R 0 = -0.001 and f /epsilon1 = g /epsilon1 = 1.2, 1.4, 1.6 respectively, following the same approach. We plotted the normalized field vectors n for these black holes in Figures (4,top;left), (4,top; middle and righr) and (4,bottom; left and right). Due to the better display of topological charges, we tried to drawtwo figures for each case, that is, Figure (4,top;left), Figures (4,top; middle and righr) together, and Figures (4,bottom; left and right) together are checked in the investigation of the total topological charges. As it is clear in these figures, only for Figure (4,top;left) corresponding to the free parameters q = 0.5, λ = 0.5, c = 0.01, R 0 = -0.001 and f /epsilon1 = g /epsilon1 = 1.2, we have topological charges ω =(-1, +1, -1) and the total topological charge W = -1. For the other two samples with f /epsilon1 = g /epsilon1 = 1.4, 1.6, the total topological charges are equal to W = 0. By comparing Figures (3) and (4), it is clear that the positive or negative of R 0 and parameter values of f /epsilon1 = g /epsilon1 play a very important role in determining the topological charges. We summarize the results in Table I.", 'B. Photon sphere': 'In our past papers [96, 97], we demonstrated that the topological photon sphere can play a fundamental and reciprocal role in interpreting the behavior of a black hole. More precisely, since the existence of a photon sphere or photon rings is one of the essential requirements of ultra-compact gravitational structures, particularly black holes[98], the topological photon sphere can serve as a crucial test for determining the parameter ranges of black holes and, consequently, their overall behavior, and vice versa. To achieve this, we first construct the regular H potential and the vector field φ [99].  \nH ( r , θ ) = √ -gtt g ϕϕ = 1 sin θ ( f ( r ) h ( r ) ) 1/2 , (55)  \nφ r = ∂ r H √ grr = √ g ( r ) ∂ r H , φ θ = ∂ θ H √ g θθ = ∂ θ H √ h ( r ) , (56)  \nwhich according to the length element and the following metric function:  \nds 2 = -dt 2 F ( r ) + dr 2 F ( r ) + ( d θ 2 + d ϕ 2 sin ( θ ) 2 ) h ( r ) , (57)  \nF ( r ) = 1 -m r + f 2 ε q 2 ( 1 + c ) r 2 + R 0 r 2 12 g 2 ε -λ q 4 f 2 ε 20 r 8 ( 1 + c ) , (58)  \nthese functions will be as follows,  \nH = √ 900 -900 m r + 900 f 2 ε q 2 ( 1 + c ) r 2 + 75 R 0 r 2 g 2 ε -45 λ q 4 f 2 ε r 8 ( 1 + c ) 30 r sin ( θ ) , (59)  \nφ r = 3 csc ( θ ) ( 2 ( -1 -c ) r 8 3 + m ( 1 + c ) r 7 -4 f 2 ε q 2 r 6 3 + λ q 4 f 2 ε 6 ) 2 r 10 ( 1 + c ) , (60)  \nφ θ = -√ 900 -900 m r + 900 f 2 ε q 2 ( 1 + c ) r 2 + 75 R 0 r 2 g 2 ε -45 λ q 4 f 2 ε r 8 ( 1 + c ) cos ( θ ) 30 r 2 sin ( θ ) 2 . (61)  \nIn this work, our objective is to investigate the impact of gravitational modifications on the effective parameters of the Euler-Heisenberg model within the context of Rainbow Gravity. Therefore, we specifically focus on the influence of parameters related to these structures. Also, given the limitation on the number of equations,  \n<!-- image -->  \nr  \n<!-- image -->  \nFIG. 3: The arrows illustrate the vector field n on a segment of the ( r h -θ ) plane for black holes in Euler-Heisenberg F(R)-Rainbow gravity, with parameters q = 0.5, λ = 0.5, c = 0.01, R 0 = + 0.001, and f /epsilon1 = g /epsilon1 = 1.2, 1.4, 1.6 respectively. The zero points (ZPs) are located within the circular loop at ( r h , θ ) . The contours (blue loop) enclose the ZPs.  \n<!-- image -->  \nFIG. 4: The arrows illustrate the vector field n on a segment of the ( r h -θ ) plane for black holes in Euler-Heisenberg F(R)-Rainbow gravity, with parameters q = 0.5, λ = 0.5, c = 0.01, R 0 = -0.001, and f /epsilon1 = g /epsilon1 = 1.2, 1.4, 1.6 respectively. The zero points (ZPs) are located within the circular loop at ( r h , θ ) . The contours (blue loop) enclose the ZPs.  \n<!-- image -->  \nwe are compelled to consider some parameters as predetermined. An interesting point to note at the outset  \nTABLE I: Summary of the results  \nis that, as can be clearly seen from equation (60), the Rainbow parameter is effectively absent in this equation. Consequently, we will concentrate our efforts on the impact of the gravitational modification parameter.', '1. Case I: f ε = 0.875 < 1': 'According to the assumed values m = 1, q = 0.5, c = 0.01, R 0 = 0.001, g ε = 1.1, we have drawn the metric function for different λ , in fig (5), the metric function has a root for λ > -1.56 and the black hole has an event horizon.  \nAccording to the method of determining topological charges [96, 97, 100] and can be inferred from Table II, the chosen λ in Figure (6) corresponds to a region where the system must possess an unstable photon sphere and a total topological charge of -1. This is confirmed by the potential function H(r) exhibiting a local maximum. Consequently, the structure will take the form of a normal black hole. However, in Figure (7), this selection pertains to a region classified as a naked singularity. As depicted in Figure (7), a global minimum appears in a space devoid of an event horizon, alongside a local maximum or the unstable photon sphere, resulting in a total topological charge of 0.', '2. Case II: f ε =1': 'In the following, in order to avoid repetition, we will only mention the Figs and tables and finally we will discuss based on the tables. The metric function for different λ ,The structure of topological charges and the diagram of the potential function H(r) of this state can be seen in Figure (8,9,10) and the parametric classifica-  \nion and general conclusions about different areas can be seen in Table III.', '3. Case III: f ε = 1.5 > 1': 'The structure of topological charges and the diagram of the potential function H(r) of this state can be seen in Figure (11) and the parametric classification and general conclusions about different areas can be seen in Table IV.', 'VII. CONCLUSION': "The study of black hole thermodynamics delves into the intriguing connections between thermodynamic properties and the topological features of black holes. This field often involves examining critical points in black hole phase diagrams and assigning topological charges to these points. A notable approach is Duan's topological current φ -mapping theory, which introduces the concept of topological charges to these critical points. This method offers a new perspective on the thermodynamic behavior of black holes and can be applied to other types of black holes, revealing more complex topological information.  \nIn conclusion, Our research has yielded several significant findings. We have successfully derived the field equations for F ( R ) -Euler-Heisenberg theory, establishing a framework to explore the interaction between modified gravity and non-linear electromagnetic effects. Additionally, we have obtained an analytical solution for a static, spherically symmetric, energy-dependent black hole with constant scalar curvature. Also, our analysis of black holes in F(R)-Euler-Heisenberg gravity's Rainbow provides significant insights into their  \nFIG. 5: Metric function with different λ for the black hole in F(R)-Euler-Heisenberg gravity's Rainbow  \n<!-- image -->  \nFIG. 6: The normal vector field n in the ( r -θ ) plane. The photon spheres are located at ( r , θ ) = ( 1.173002072, 1.57 ) with respect to ( λ = -1, q = 0.5, m = 1, c = 0.01, R 0 = 0.001, g ε = 1.1 ) , Right fig is the topological potential H(r) for black hole in F(R)-Euler-Heisenberg gravity's Rainbow  \n<!-- image -->  \n<!-- image -->  \nFIG. 7: The normal vector field n in the ( r -θ ) plane. The photon spheres are located at ( r , θ ) = ( 0.9530605080, 1.57 ) , ( r , θ ) = ( 1.102139574, 1.57 ) with respect to ( λ = -9, q = 0.5, m = 1, c = 0.01, R 0 = 0.001, g ε = 1.1 ) , Right fig is the topological potential H(r) for black hole in F(R)-Euler-Heisenberg gravity's Rainbow  \n<!-- image -->  \ntopological properties. By examining the normalized field lines and various free parameters, we identified distinct winding and total topological numbers. Our findings suggest that the parameters ( R 0 ) and ( f /epsilon1 = g /epsilon1 ) influence the topological charges. The stability of  \nthese black holes was further assessed through winding numbers, highlighting the intricate relationship between these parameters and the black holes' topological characteristics. These results are comprehensively summarized in Table I. Additionally, an overview of Tables  \nTABLE II: ∗ Unauthorized region: is the region where the roots of φ equations become negative or imaginary in this region. TTC: ∗ Total Topological Charge. R PLPS : ∗ The minimum or maximum possible radius for the appearance of an unstable photon sphere.  \nFIG. 8: Metric function with different λ for the black hole in F(R)-Euler-Heisenberg gravity's Rainbow  \n<!-- image -->  \n<!-- image -->  \nFIG. 9: The normal vector field n in the ( r -θ ) plane. The photon spheres are located at ( r , θ ) = ( 1.033146077, 1.57 ) respect to with respect to ( λ = 1, q = 0.5, m = 1, c = 0.01, R 0 = 0.001, g ε = 1.1 ) , Right fig is the topological potential H(r) for Euler-Heisenberg black hole model  \n<!-- image -->  \nII, III, and IV related to the photon sphere of the mentioned black hole shows that with an increase in f ε , the permissible range of negative λ in the first case gradually transitions into a non-permissible region in the third  \ncase. Furthermore, the QED parameter, which measures the strength of nonlinear effects, can be either positive or negative. A positive QED parameter reduces the electric field near the horizon and increases the black hole's  \nFIG. 11: The normal vector field n in the ( r -θ ) plane. The photon spheres are located at ( r , θ ) = ( 0.5644005040, 1.57 ) with respect to with respect to ( λ = 0.544, q = 0.5, m = 1, c = 0.01, R 0 = 0.001, g ε = 1.1 ) F(R)-Euler-Heisenberg gravity's Rainbow  \n<!-- image -->  \nFIG. 10: The normal vector field n in the ( r -θ ) plane. The photon spheres are located at ( r , θ ) = ( 0.7097591462, 1.57 ) , ( r , θ ) = ( 0.9921280053, 1.57 ) with respect to  \n<!-- image -->  \n( λ = -0.544, q = 0.5, m = 1, c = 0.01, R 0 = 0.001, g ε = 1.1 ) , Right fig is the topological potential H(r) for black hole in F(R)-Euler-Heisenberg gravity's RainbowTABLE III: *Unauthorized region: is the region where the roots of φ equations become negative or imaginary in this region. TTC: *Total Topological Charge. 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