  The thermodynamics and the entanglement properties of two-dimensional
conformal field theories ($2$d CFTs) on curved backgrounds are studied. By
means of conformal mapping we study the equivalent system on flat space
governed by the deformed Hamiltonian, which is a spatial integral of the
Hamiltonian density modulated by an enveloping function. Focusing on
holographic CFTs, we observe Hawking-Page like phase transition for the thermal
and the entanglement entropy as we vary the background metric. We also compute
the mutual information to study the information theoretic correlation between
parts of the curved spacetime. The gravity dual of 2d CFTs on curved background
is also discussed.
