We obtain renormalized stress tensor of a mass-less, charge-less dynamical quantum scalar field, minimally coupled with a spherically symmetric static Lukewarm black hole. In two dimensional analog the minimal coupling reduces to the conformal coupling and the stress tensor is found to be determined by the nonlocal contribution of the anomalous trace and some additional parameters in close relation to the work presented by Christensen and Fulling. Lukewarm black holes are a special class of Reissner-Nordström-de Sitter space times where its electric charge is equal to its mass. Having the obtained renormalized stress tensor we attempt to obtain a time-independent solution of the well known metric back reaction equation. Mathematical derivations predict that the final state of an evaporating quantum Lukewarm black hole reduces to a remnant stable mini black hole with moved locations of the horizons. Namely the perturbed black hole (cosmological) horizon is compressed (extended) to scales which is smaller (larger) than the corresponding classical radius of the event horizons. Hence there is not obtained an deviation on the cosmic sensor-ship hypothesis.

Aim of the paper is study weak gravitational lensing of quantum (perturbed) (QLBHL) and classical (CLBHL) Lukewarm black hole in presence of cosmological parameter Λ. We apply numerical method to evaluate deflection angle of bending light rays, images locations θ of sample source β = − π 4 , and corresponding magnifications µ. There is not obtained real values for Einstein rings locations θ E (β = 0) for CLBHL but we calculate them for QLBHL. As experimental test of our calculations, we choose mass M of 60 type of most massive observed galactic black holes as gravitational lens and study quantum matter fields effects on the angle of bending light rays in presence of the cosmological constant effects. We calculate locations of non-relativistic images and corresponding magnifications. Numerical diagrams show that the quantum matter effects cause to be reduce absolute values of the quantum deflection angle with respect to the classical ones. Sign of the quantum deflection angle is changed with respect to the classical values in presence of the cosmological constant. This means dominance of anti-gravity counterpart of the cosmological horizon on the bending light rays angle with respect to absorbing effects of local 60 type of observed most massive black holes. Variations of the image positions and magnifications are negligible by increasing dimensionless cosmological constant ǫ = 16ΛM 2 3. Deflection angle takes positive (negative) values for CLBHL (QLBHL) and they decrease very fast (slow) by increasing closest distance x 0 of bending light ray and/or dimensionless cosmological parameter for sample giant black holes 0.001 < ǫ < 0.01.

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