I t h a s b e e n s h own th a t Ad S / C F T c o r r e s p o n d e n c e ma y b e r e a l i z e d b e t w e e n s o m e A d S - l i ke c o s m o l o g i c a l s p a c e a n d CF T l iv i n g o n t h e b o u n d a r y. By ex t e n d i n g s u c h wo r k s we s t u d i e d the role o f R2 term in the corrected action to the thermodynamic quantities, particularly by adding a boundary counterterms to the gravitational action and considering an AdS-like cosmological space. With such HD (higher derivative) terms and metric we calculated the free energy, as a function of N2T4 times 3/2(1−1/N2), that is comparable with perturbative result followed from boundary QFT. In fact the difference between the results obtained from AdS-like cosmological space and the results obtained from strong coupling limit of ℵ = 4 SCFT, by taking into account next to leading term in large N expansion, is just a 3/2 factor. Higher derivative term contributions also appear in entropy and energy via the redefinition of gravitation constant. We explicitly identify higher order counterterms effect in energy as a constant term namely E0. This value can be interpreted as the Casimir energy, so that, there is a full and perfect match between QFT and gravitational action when both R2 term and boundary counterterms are considered. So we can claim that our results have assured us that there is a good fit between AdS-like cosmological space and ℵ = 4 SU(N) super Yang-Mills quantum theory

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