  We present an analytic, perturbative solution to the Einstein equations with
a scalar field that describes dynamical black holes in a slow-roll inflationary
cosmology. We show that the metric evolves quasi-statically through a sequence
of Schwarzschild-de Sitter like metrics with time dependent cosmological
constant and mass parameters, such that the cosmological constant is
instantaneously equal to the value of the scalar potential. The areas of the
black hole and cosmological horizons each increase in time as the effective
cosmological constant decreases, and the fractional area increase is
proportional to the fractional change of the cosmological constant, times a
geometrical factor. For black holes ranging in size from much smaller than to
comparable to the cosmological horizon, the pre-factor varies from very small
to order one. The "mass first law" and the "Schwarzschild-de Sitter patch first
law" of thermodynamics are satisfied throughout the evolution.

High Energy Physics - Theory;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology