Searching for dynamical black holes in various theories of gravity

We construct models of Einstein and $f(R)$ gravity with two scalar fields, which admit analytical solutions describing time-varying dynamical black holes. Their thermodynamics is investigated in the adiabatic approximation. In addition to Misner-Sharp-Hernandez quasilocal mass, we provide time-dependent thermodynamical quantities, including Hawking temperature, Helmholtz free energy, entropy, energy. The latter does not always coincide mass at horizon, although they static limit. For Schwarzschild-type (i.e., ${g}_{tt}{g}_{rr}=\ensuremath{-}1$) holes gravity, one scalars a ghost negative kinetic show that this can be avoided gravity.