  We study the evolution of horizons of black holes in the $1+1+2$ covariant
setting and investigate various properties intrinsic to the geometry of the
foliation surfaces of these horizons. This is done by interpreting formulations
of various quantities in terms of the geometric and thermodynamic quantities.
We establish a causal classification for horizons in different classes of
spacetimes. We have also recovered results by Ben-Dov and Senovilla which put
cut-offs on the equation of state parameter $\sigma$, determining the
spacelike, timelike and non-expanding horizons in the the Robertson-Walker
class of spacetimes. We show that stability of marginally trapped surfaces
(MTS) in the Robertson-Walker spacetimes is only achievable under the
conditions of negative pressure, and also classify the spacelike future outer
trapping horizons (SFOTH) in the Robertson-Walker spacetimes via bounds on the
equation of state parameter $\sigma$. For the Lemaitre-Tolman-Bondi (LTB)
model, it is shown that a relationship between the energy density and the
electric part of the Weyl curvature, $\mathcal{E}$, gives the causal
classification of the MTTs. It is further shown that only spacelike MTTs are
foliated by stable MTS, and that this stability guarantees no shell crossing.
We also provide an explicit proof of the third law of black hole thermodynamics
for the LRS II class of spacetimes, and by extension, any spacetime whose
outgoing and ingoing null geodesics are normal to the MTS.
