August 26, 2016 02:45 PM

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Babies are known to be early risers, but new dad Adam Brody doesn’t seem to mind one bit.

“I do a nice morning hike with my daughter,” The O.C. alum, 36, tells PEOPLE exclusively of his 1-year-old little girl Arlo Day with musician and actress wife Leighton Meester. “That’s always a treat.”

Continues Brody, “It’s something to do in the wee hours of the morning when she’s up. It’s a good activity for both of us.”

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“We hike around our house,” the actor adds. “Watching the sunrise with a kid is pretty fun. I have her in the [Baby] Bjorn.”

Brody wed Meester, 30, in an ultra-private ceremony in early 2014, one year after they began dating. And their daughter’s birth was kept under wraps, as well, with confirmation coming more than a month after she arrived.

The couple were recently snapped out at the premiere for Brody’s new Crackle series StartUp, which debuts Sept. 6.

Brody shares that he likes to spend time building “the kids stuff” around his house, like Arlo’s slide and rocking chair.

But he also admits he suffered an injury recently — spraining his MCL while surfing — that makes keeping up with Arlo a little more difficult than usual.

“Three weeks later, I’m still limping. I can bend it, but it hurts,” he says. “Chasing the kid is pretty hard! It’s put a big monkey wrench in that.”

For more from Adam Brody, pick up the latest issue of PEOPLE, on newsstands now.

Aili Nahas and Jen Juneau

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