This paper presents a derivation of holographic entanglement entropy for spherical entangling surfaces. The authors begin by mapping the boundary conformal field theory (CFT) to a hyperbolic geometry, where the vacuum state is transformed into a thermal state. This mapping allows the entanglement entropy to be interpreted as the thermodynamic entropy of the thermal state. Using the AdS/CFT correspondence, the thermodynamic entropy is calculated as the horizon entropy of a topological black hole. In even dimensions, the universal contribution to the entanglement entropy is shown to be given by the A-type trace anomaly for any CFT, without reference to holography. The paper discusses the entanglement entropy in flat space and in a cylindrical background geometry. In flat space, the entanglement entropy is calculated using the modular Hamiltonian, which is shown to be a local operator in the causal development of the region. The vacuum correlators in this region are conformally mapped to thermal correlators in a hyperbolic space, leading to a thermal density matrix. This result is extended to a cylindrical background geometry, where the same approach is applied. In the AdS story, the authors show that the thermal state in the boundary CFT is dual to a black hole in the bulk gravity theory. The entanglement entropy is then calculated as the horizon entropy of the black hole. The authors also demonstrate that the universal contribution to the entanglement entropy in even dimensions is given by the A-type trace anomaly. The paper concludes by showing that the results can be applied to any gravitational theory, not just Einstein gravity. The results are consistent with the Ryu-Takayanagi conjecture for holographic entanglement entropy.This paper presents a derivation of holographic entanglement entropy for spherical entangling surfaces. The authors begin by mapping the boundary conformal field theory (CFT) to a hyperbolic geometry, where the vacuum state is transformed into a thermal state. This mapping allows the entanglement entropy to be interpreted as the thermodynamic entropy of the thermal state. Using the AdS/CFT correspondence, the thermodynamic entropy is calculated as the horizon entropy of a topological black hole. In even dimensions, the universal contribution to the entanglement entropy is shown to be given by the A-type trace anomaly for any CFT, without reference to holography. The paper discusses the entanglement entropy in flat space and in a cylindrical background geometry. In flat space, the entanglement entropy is calculated using the modular Hamiltonian, which is shown to be a local operator in the causal development of the region. The vacuum correlators in this region are conformally mapped to thermal correlators in a hyperbolic space, leading to a thermal density matrix. This result is extended to a cylindrical background geometry, where the same approach is applied. In the AdS story, the authors show that the thermal state in the boundary CFT is dual to a black hole in the bulk gravity theory. The entanglement entropy is then calculated as the horizon entropy of the black hole. The authors also demonstrate that the universal contribution to the entanglement entropy in even dimensions is given by the A-type trace anomaly. The paper concludes by showing that the results can be applied to any gravitational theory, not just Einstein gravity. The results are consistent with the Ryu-Takayanagi conjecture for holographic entanglement entropy.