  The importance of the first-class constraint algebra of general relativity is
not limited just by its self-contained description of the gauge nature of
spacetime, but it also provides conditions to properly evolve the geometry by
selecting a gauge only once throughout the whole evolution of a gravitational
system. This must be a property of all background independent theories. In this
paper we consider gravitational theories which arise from deformations of the
fundamental canonical variables of general relativity where the proposed
deformations are inspired by modifications of gravity. These variable
deformations result in new theories when the deformation is not a canonical
transformation. The new theory must preserve the first-class structure of the
algebra, which is a non-trivial restriction for generic deformations. In this
vein we present a general deformation scheme along with consistency conditions,
so that the algebra of constraints is still satisfied in the resulting theory.
This is illustrated both in metric theory as well as in tetrad theory.
