ERJ 145 First Officer reported a battery overheat condition at the gate. Maintenance replaced the battery and the aircraft departed. In cruise; the replacement battery began to overheat and flight crew continued to destination where maintenance was requested.

Narrative

At the gate pre departure; we had received an EICAS message for BATT 2 OVERHEAT. We accomplished memory items and ran the associated QRH checklist. The battery initially stabilized at 70-73 degrees Celsius. We contacted maintenance control and deplaned the aircraft. We called operations to have crash fire and rescue nearby. While deplaning the aircraft after approximately 5 minutes the battery rose to 81 degrees Celsius. At this time we turned all power off of the airplane and the crew also got off the ship. We did not [request priority handling] because we were on the gate. Maintenance swapped out the battery and closed the written up maintenance item.We boarded up the plane and departed to ZZZ. As we were taxiing out our Battery 2 was 36 degrees Celsius but well within limits. As we departed ZZZ; on the climb out our battery temperature rose to 40 degrees and continued to slowly rise until reaching 58 degrees. The battery temperature stabilized at 58 degrees for approximately 15 minutes until we began our descent into ZZZ1. We landed the plane in ZZZ1 with a battery temperature of approximately 66 degrees. The battery temperature rose by approximately 2 degrees per minute from 58 through 70 degrees when the Master Warning went off. We were taxiing from Runway XR to gate XX and just before crossing XL our Master Warning went off. We ran through the QRH memory items and again the QRH for BATT 2 OVERTEMP. We [requested priority] to the gate and wanted crash fire and rescue to again be there standing by. Upon reaching the gate we turned all the power off the plane and called maintenance to write it up.Continue training on QRH memory items for pilots. When a maintenance issue is found; Maintenance should do more work to find the root of the problem instead of finding the easiest possible solution. I don't feel like Maintenance Control did their due diligence to find the root of the problem and put people at risk unnecessarily.