Temp and time for using half the recipe of cake
As @joe said, it does not scale linearly. 

There are various factors that effects cooking time: 

Thickness: the heat transfers mainly from the top and bottom for a round on a not too thick to thin dish, so for half a cake with the same thickness the time will stay about the same. Think of a pizza for example, whether it's 5 or 15 inches in diameter the cooking time is nearly exactly the same. For a very thick cake, though (sponge cake for a wedding cake for example), the lateral heat transfer will matter far more. 

Chemical reactions: Some chemical reactions (egg coagulation, Maillard, unfolding of proteins etc.) happens at a very specific temp under which they don't happen at all or takes forever. For example  the  coagulation an egg at ambient temperature will take several years, where it would happen in a few minutes at 65 centigrades. Try browning a toast at 90 c (half of the temp needed), it'll never happens. It takes 180c to start burning bread, anything under will stop the expected result from happening .  

Most chemical or physical reactions (water boiling) that gives the expected taste or texture to something you bake happens between 60 to 180 c so do not change the temp when changing the amount in a recipe if the temp indicated is within that range. 

If you cook a bigger item a higher temperature it won't have a chance to cook internally before being burnt on the outside, on the opposite, a smaller item with the same shape will be crisp and nice at the same temp. 

As for the time needed, only on very compact items, nearly spherical, such as scones, choux pastry, or poultry (think of the time needed to roast a breast of chicken vs a whole chicken vs a turkey) the amount of time will be more closely proportional to the size or weight. You cook the smaller item less longer. 

In brief: don't change the temp if it's equal or less than 180c. If your cake is compact (like a banana bread cooked in a tin) drop maybe  around a quarter  of the time for a halving of the recipe, and maybe a sixth of the time for a thinner cake.