  In this note we explore the application of modular invariance in
2-dimensional CFT to derive universal bounds for quantities describing certain
state degeneracies, such as the thermodynamic entropy, or the number of
marginal operators. We show that the entropy at inverse temperature 2 pi
satisfies a universal lower bound, and we enumerate the principal obstacles to
deriving upper bounds on entropies or quantum mechanical degeneracies for fully
general CFTs. We then restrict our attention to infrared stable CFT with
moderately low central charge, in addition to the usual assumptions of modular
invariance, unitarity and discrete operator spectrum. For CFT in the range
c_left + c_right < 48 with no relevant operators, we are able to prove an upper
bound on the thermodynamic entropy at inverse temperature 2 pi. Under the same
conditions we also prove that a CFT can have a number of marginal deformations
no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2.
