  Markovian master equations provide a versatile tool for describing open
quantum systems when memory effects of the environment may be neglected. As
these equations are of an approximate nature, they often do not respect the
laws of thermodynamics when no secular approximation is performed in their
derivation. Here we introduce a Markovian master equation that is
thermodynamically consistent and provides an accurate description whenever
memory effects can be neglected. The thermodynamic consistency is obtained
through a rescaled Hamiltonian for the thermodynamic bookkeeping, exploiting
the fact that a Markovian description implies a limited resolution for heat.
Our results enable a thermodynamically consistent description of a variety of
systems where the secular approximation breaks down.
