We argue that the equations of motion of quantum field theories in curved backgrounds encode new fundamental black hole thermodynamic relations. We define new entropy variation relations. These `emerge through the monodromies that capture the infinitesimal changes in the black hole background produced by the field excitations. This raises the possibility of new thermodynamic relations defined as independent sums involving entropies, temperatures and angular velocities defined at every black hole horizon. We present explicit results for the sum of all horizon entropy variations for general rotating black holes, both in asymptotically at and asymptotically anti-de Sitter spacetimes in four and higher dimensions. The expressions are universal, and in most cases add up to zero. We also find that these thermodynamic summation relations apply in theories involving multi-charge black holes.
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