  Given a hyperbolic homeomorphism on a compact metric space, consider the
space of linear cocycles over this base dynamics which are H\"older continuous
and whose projective actions are partially hyperbolic dynamical systems. We
prove that locally near any typical cocycle, the Lyapunov exponents are
H\"older continuous functions relative to the uniform topology. This result is
obtained as a consequence of a uniform large deviations type estimate in the
space of cocycles. As a byproduct of our approach, we also establish other
statistical properties for the iterates of such cocycles, namely a central
limit theorem and a large deviations principle.
