  By looking at the Lovelock theorem one can infer that the gravity model given
by \cite{GL} could not applicable for all type of 4D Einstein Gauss Bonnet (GB)
curved spacetimes. Because in 4D space time the Gauss-Bonnet invariant is a
total derivative, and hence it does not contribute to the gravitational
dynamics. Hence, authors in the work \cite{AL} presented an alternative
consistent EGB gravity model instead of \cite{GL} by applying break of
diffeomorphism property. In this work we use it to produce a dS/AdS black hole
metric and then investigate its thermodynamic behavior in presence of cloud of
Numbo Goto strings.
