We used that route and have had snow in April in New Mexico and Arizona.On the 3rd day we were in Amarillo, Texas and it was 90. The next night in Gallup, NM it was 15 and snowing. The next day we were in Wlliams, AZ and it was 32 with about 6" of snow on the ground.
For me a better route is Pa Turnpike to I-70 to I-44 in Missouri and then pick up I-40 in Oklahoma. Going I-80 south you may be more likely to hit snow or ice at some of the higher elevations. As others have said,watch the weather and be prepared to lay over a day or two.