  Pulsar Timing Array (PTA) collaborations recently reported evidence for the
presence of a gravitational wave background (GWB) in their datasets. The main
candidate that is expected to produce such a GWB is the population of
supermassive black hole binaries (SMBHB). Some analyses showed that the
recovered signal may exhibit time-dependent properties, i.e. non-stationarity.
In this paper, we propose an approximated non-stationary Gaussian process (GP)
model obtained from the perturbation of stationary processes. The presented
method is applied to the second data release of the European pulsar timing
array to search for non-stationary features in the GWB. We analyzed the data in
different time slices and showed that the inferred properties of the GWB evolve
with time. We find no evidence for such non-stationary behavior and the Bayes
factor in favor of the latter is $\mathcal{B}^{NS}_{S} = 1.5$. We argue that
the evolution of the GWB properties most likely comes from the \mf{improvement
of the observation cadence} with time and \mf{better} characterization of the
noise of individual pulsars. Such non-stationary GWB could also be produced by
the leakage of non-stationary features in the noise of individual pulsars or by
the presence of an eccentric single source.
