  We give a method to bound the entropy of measures on
$SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ which are invariant under a one
parameter diagonal subgroup, in terms of entropy contributions from the regions
of the cusp corresponding to different parabolic groups. These bounds depend on
an auxiliary linear functional on the Lie algebra of the Cartan group. In
follow-up papers we will show how to optimize this functional to get good
bounds on the cusp entropy and prove that in many cases these bounds are sharp.
