  Contrary to Earth, which has a small orbital eccentricity, some exoplanets
discovered in the insolation habitable zone (HZ) have high orbital
eccentricities (e.g., up to an eccentricity of $\sim0.97$ for HD~20782~b). This
raises the question of the capacity of these planets to host surface liquid
water. In order to assess the habitability of an eccentric planet, the mean
flux approximation is often used. It states that a planet on an eccentric orbit
is called habitable if it receives on average a flux compatible with the
presence of surface liquid water. However, as the planets do experience
important insolation variations over one orbit and even spend some time outside
the HZ for high eccentricities, the question of their habitability might not be
as straightforward. We performed a set of simulations using the Global Climate
Model LMDz, exploring the limits of the mean flux approximation when varying
the luminosity of the host star and the eccentricity of the planet. We computed
the climate of tidally locked ocean covered planets with orbital eccentricity
from 0 to 0.9 receiving a mean flux equal to Earth's, around stars of
luminosity ranging from $L_\odot$ to $10^{-4}~L_\odot$. Using here a definition
of habitability based on the presence of surface liquid water, we find that
most of the planets considered can sustain surface liquid water on the dayside
with an ice cap on the nightside. However, for high eccentricity and high
luminosity, planets cannot sustain surface liquid water during the whole
orbital period. They completely freeze at apoastron and when approaching
periastron an ocean appears around the substellar point. We conclude that the
higher the eccentricity and the higher the luminosity of the star, the less
reliable the mean flux approximation.
