Although they are both very similar, I have to say I don't like 'bear' very much in this context but I can't really say why. I think it's that you can't really 'bear' the summer - and as everyone has said, it's not the summer that needs to be borne, it's the heat, so I would suggest the use of either 'endure' or even 'survive'.
He had to endure the heat of the summer
or He had to survive the summer. (in other words, get to the end of it)
I would tend to use 'bear' more in:
He couldn't bear the heat of the summer.
I couldn't bear it!
He couldn't bear to see them leave.
We just had to grin and bear it.
He found the heat of the summer unbearable.