Can you start a cake at 350 F (180 C), then lower the oven to 325 F (160 C)? What happens to cake? Recipe calls for 350 but the recipe is old. Other recipes I have checked online say to bake at a lower temp of 325.
You can, but I cannot think of a single good reason why you would.
325 F and 350 F are not that far apart; they may even be within the accuracy of most ovens, especially older ones.
When you bake the cake, the two main modalities of heat transfer (radiation and conduction from the air in contact with the surface of the cake) begin to heat up the surface of the cake at the cake/air or cake/pan interface.
Within the body of the batter, conduction begins to transfer heat into the center of the cake.
At the same time, on the top surface especially, water can evaporate, faster than it is replenished from water diffusing from the interior of the batter. This allows crust formation to begin, with drying of the surface, and heating of it to levels above 250 F or so which allow caramelization and Maillard reactions to begin in the crust.
The interior of the cake can never exceed the boiling point of water, and so browning does not happen.
Thus, the higher the temperature, the greater the difference between the surface temperature of the cake as baking proceeds, and the interior of the cake. This has several consequences:
However, 25 F is not enough of a difference to worry about in most cases.
Starting at only a slightly higher temperature, when the water from the surface is not yet evaporated, will have only a minimal effect on any of this. There is little, if any point to doing so.