  The resource theory of thermal operations explains the state transformations
that are possible in a very specific thermodynamic setting: there is only one
thermal bath, auxiliary systems can only be in corresponding thermal state
(free states), and the interaction must commute with the free Hamiltonian (free
operation). In this paper we study the mildest deviation: the reservoir
particles are subject to inhomogeneities, either in the local temperature
(introducing resource states) or in the local Hamiltonian (generating a
resource operation). For small inhomogeneities, the two models generate the
same channel and thus the same state transformations. However, their
thermodynamics is significantly different when it comes to work generation or
to the interpretation of the "second laws of thermal operations".
