And don't worry if your HR is higher. When I run, I'm always in the high 160's (which is far too high according to norms). I'm not breathing hard, not struggling- but it's just where I run. My HR shoots up so high on the run, I cannot keep it down. It'll be in the 130's when I walk, then I run and it shoots up to the 160's. My resting HR is in the 50's so it's not like my heart is out of conditioning or anything.
I stressed about it for a long time, but finally realized that it is what it is. Last year I did some HR training and actually got it to stay in the 150's, but that's the lowest I've EVER gotten it to stay during a run.
Some people just have a higher HR when working out, and that's OK, too.