A higher price will likely result in more mining activity, but not more bitcoins being mined. Miners could sell a smaller portion of their mined coins to meet expenses, allowing them to keep more for themselves. Basically it's a much less liquid market, one reason why it is so much more volatile.
Those two statements don't go together. Higher bitcoin price = more mining activity = more costs per bitcoin. It will always level out, up or down. That is the beauty of mining.