Change in current in an electrolytic cell over time
The electrolyte conductivity is dependent on the temperature of the electrolyte. Higher temperature increases ions mobility, thereby increasing electrolyte conductivity. The conductivity increases around 2% per degree (assuming linear increase of conductivity). More info here.

As Burak Ulgut mentioned, temperature increase is due to ohmic (Joule) heating. This can be estimated accordingly: $Q=V \cdot I \cdot t$