  Plate tectonics is a fundamental component for the habitability of the Earth.
Yet whether it is a recurrent feature of terrestrial bodies orbiting other
stars or unique to the Earth is unknown. The stagnant lid may rather be the
most common tectonic expression on such bodies. To understand whether a
stagnant-lid planet can be habitable, i.e. host liquid water at its surface, we
model the thermal evolution of the mantle, volcanic outgassing of H$_2$O and
CO$_2$, and resulting climate of an Earth-like planet lacking plate tectonics.
We used a 1D model of parameterized convection to simulate the evolution of
melt generation and the build-up of an atmosphere of H$_2$O and CO$_2$ over 4.5
Gyr. We then employed a 1D radiative-convective atmosphere model to calculate
the global mean atmospheric temperature and the boundaries of the habitable
zone (HZ). The evolution of the interior is characterized by the initial
production of a large amount of partial melt accompanied by a rapid outgassing
of H$_2$O and CO$_2$. At 1 au, the obtained temperatures generally allow for
liquid water on the surface nearly over the entire evolution. While the outer
edge of the HZ is mostly influenced by the amount of outgassed CO$_2$, the
inner edge presents a more complex behaviour that is dependent on the partial
pressures of both gases. At 1 au, the stagnant-lid planet considered would be
regarded as habitable. The width of the HZ at the end of the evolution, albeit
influenced by the amount of outgassed CO$_2$, can vary in a non-monotonic way
depending on the extent of the outgassed H$_2$O reservoir. Our results suggest
that stagnant-lid planets can be habitable over geological timescales and that
joint modelling of interior evolution, volcanic outgassing, and accompanying
climate is necessary to robustly characterize planetary habitability.
