It depends on the resistivity of the electrode materials. If they are different there will be a difference in the extent they heat up. In general electrodes are made of low resistivity materials to minimise such losses. Resistance is equal to resistivity x length / area. If resistivity is constant and you double the cross sectional area and length of one electrode the resistance effectively remains the same.
I think heat loss during the electrolysis can be explain on the basis of “Joule heating” or "Ohm's law". The most general and fundamental formula for Joule heating is:
P = (Va-Vb) I
Where P is power or heat loss, I is applied current and Va-Vb is the voltage drop across the electrode. Electrical resistances and transport resistance lead to energy waste in the form of heat generation. Greater the area or length of electrode, greater will be the resistance, greater will be the heat loss.
This link can help you with more explanations:
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-40422013000800017
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