  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.
