We started half birthdays when my daughter was six months not because it was hot or anything but because it is fun to do. At six months old, I cut a cupcake in half and put half a candle on it. We sang and blew out the candle for her. We took photos of course. My husband and I laughed and thought it was so cute and then hubby and I shared the cupcake together since my daughter was too young to eat it just yet. We decided to make it a yearly family tradition after the six month celebration just because we enjoyed it so much.
At one and half, I put one full candle and half of another in a cupcake. We did the same as the six month half party but my daughter ate the cupcake this time. At two and a half, I got two mini cupcakes and put a candle in each cupcake and a half of one in one of them. At three and four just had one cupcake and put the candles in it.
My daughter is 5 now and every year she looks forward to her half birthday almost as much as her real birthday. We do not do presents and only invite maybe two friends over and they just play. We only do cupcakes and ice cream. At her four and half, I bought some Tangled party favors at the dollar store (Dollar Tree) and she loved that!
Maybe you can go to the zoo in the Fall like Sept when it is cooler and just call it a half birthday? I know January is really her half year mark but January would probably be too cold.